6.6.2 2.2

6.6.2.1 [1388] Problem 1

problem number 1388

Added April 14, 2019.

Problem Chapter 6.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 x y+b_2 x^2+c_2 x) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*x*y+b2*x^2+c2*x)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-2 \text {a1} \text {b2} x^3-3 \text {a1} \text {c2} x^2+6 \text {a1} z+2 \text {a2} \text {b1} x^3+3 \text {a2} \text {c1} x^2-6 \text {a2} y}{6 \text {a1}},\frac {e^{-\frac {\text {a1} x^2}{2}} (\text {a1} y+\text {b1} x+\text {c1})}{\text {a1}}-\frac {\sqrt {\frac {\pi }{2}} \text {b1} \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\text {a1}^{3/2}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*x*y+b2*x^2+c2*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{-\frac {\operatorname {a1} \,x^{2}}{2}} \sqrt {\pi }\, \sqrt {\operatorname {a1}}\, \left (\operatorname {a1} y +\operatorname {b1} x +\operatorname {c1} \right )-\frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\operatorname {a1}}\, x}{2}\right ) \sqrt {2}\, \pi \operatorname {b1}}{2}}{\sqrt {\pi }\, \operatorname {a1}^{{3}/{2}}}, \frac {\left (-2 \operatorname {b2} \,x^{3}-3 \operatorname {c2} \,x^{2}+6 z \right ) \operatorname {a1}^{2}+2 \operatorname {a2} \left (\operatorname {b1} \,x^{3}+\frac {3}{2} \operatorname {c1} \,x^{2}-3 y \right ) \operatorname {a1} -6 \operatorname {a2} \operatorname {c1}}{6 \operatorname {a1}^{2}}\right )\]

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6.6.2.2 [1389] Problem 2

problem number 1389

Added April 14, 2019.

Problem Chapter 6.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 x z+b_2 x^2+c_2 x) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*x*z+b2*x^2+c2*x)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\frac {\text {a1} x^2}{2}} (\text {a1} y+\text {b1} x+\text {c1})}{\text {a1}}-\frac {\sqrt {\frac {\pi }{2}} \text {b1} \text {erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\text {a1}^{3/2}},\frac {e^{-\frac {\text {a2} x^2}{2}} (\text {a2} z+\text {b2} x+\text {c2})}{\text {a2}}-\frac {\sqrt {\frac {\pi }{2}} \text {b2} \text {erf}\left (\frac {\sqrt {\text {a2}} x}{\sqrt {2}}\right )}{\text {a2}^{3/2}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*x*z+b2*x^2+c2*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {{\mathrm e}^{-\frac {\operatorname {a1} \,x^{2}}{2}} \sqrt {\pi }\, \sqrt {\operatorname {a1}}\, \left (\operatorname {a1} y +\operatorname {b1} x +\operatorname {c1} \right )-\frac {\sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\operatorname {a1}}\, x}{2}\right ) \pi \operatorname {b1}}{2}}{\operatorname {a1}^{{3}/{2}} \sqrt {\pi }}, \frac {{\mathrm e}^{-\frac {\operatorname {a2} \,x^{2}}{2}} \sqrt {\pi }\, \sqrt {\operatorname {a2}}\, \left (\operatorname {a2} z +\operatorname {b2} x +\operatorname {c2} \right )-\frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\operatorname {a2}}\, x}{2}\right ) \sqrt {2}\, \pi \operatorname {b2}}{2}}{\operatorname {a2}^{{3}/{2}} \sqrt {\pi }}\right )\]

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6.6.2.3 [1390] Problem 3

problem number 1390

Added April 14, 2019.

Problem Chapter 6.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x +(a_1 x y+b_1 x^2+c_1 x) w_y + (a_2 y z+b_2 y^2+c_2 y) w_z= 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =   D[w[x, y,z], x] +(a1*x*y+b1*x^2+c1*x)*D[w[x, y,z], y] +(a2*y*z+b2*y^2+c2*y)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$Aborted

Maple ✓

restart; 
pde :=   diff(w(x,y,z),x)+(a1*x*y+b1*x^2+c1*x)*diff(w(x,y,z),y)+ (a2*y*z+b2*y^2+c2*y)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[\text {Expression too large to display}\]

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6.6.2.4 [1391] Problem 4

problem number 1391

Added April 14, 2019.

Problem Chapter 6.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x +(a_1 x y+b_1 y^2) w_y + (a_2 x z+b_2 z^2) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   D[w[x, y,z], x] +(a1*x+b1*y^2)*D[w[x, y,z], y] +(a2*x*z+b2*z^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {2 \left (\text {b1} x y \operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )+\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )\right )}{(2 \text {b1} x y+1) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )+\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )-\sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\text {a1}} \sqrt {\text {b1}} x^{3/2}\right )},\frac {\sqrt {\frac {\pi }{2}} \text {b2} \text {erfi}\left (\frac {\sqrt {\text {a2}} x}{\sqrt {2}}\right )}{\sqrt {\text {a2}}}+\frac {e^{\frac {\text {a2} x^2}{2}}}{z}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   diff(w(x,y,z),x)+(a1*x+b1*y^2)*diff(w(x,y,z),y)+ (a2*x*z+b2*z^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-\operatorname {AiryBi}\left (-\left (\operatorname {b1} \operatorname {a1} \right )^{{1}/{3}} x \right ) y \operatorname {b1} +\left (\operatorname {b1} \operatorname {a1} \right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -\left (\operatorname {b1} \operatorname {a1} \right )^{{1}/{3}} x \right )}{\operatorname {AiryAi}\left (-\left (\operatorname {b1} \operatorname {a1} \right )^{{1}/{3}} x \right ) y \operatorname {b1} -\left (\operatorname {b1} \operatorname {a1} \right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -\left (\operatorname {b1} \operatorname {a1} \right )^{{1}/{3}} x \right )}, \frac {\left (\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-\operatorname {a2}}\, x}{2}\right ) z \operatorname {b2} +\sqrt {2}\, \sqrt {-\operatorname {a2}}\, {\mathrm e}^{\frac {\operatorname {a2} \,x^{2}}{2}}\right ) \sqrt {2}}{2 \sqrt {-\operatorname {a2}}\, z}\right )\]

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6.6.2.5 [1392] Problem 5

problem number 1392

Added April 14, 2019.

Problem Chapter 6.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + x z w_y - x y w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   a*D[w[x, y,z], x] +x*z*D[w[x, y,z], y] -x*y*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y \sin \left (\frac {x^2}{2 a}\right )+z \cos \left (\frac {x^2}{2 a}\right ),y \cos \left (\frac {x^2}{2 a}\right )-z \sin \left (\frac {x^2}{2 a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   a*diff(w(x,y,z),x)+x*z*diff(w(x,y,z),y)- x*y*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y^{2}+z^{2}, -2 a \arctan \left (\frac {y}{z}\right )+x^{2}\right )\]

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6.6.2.6 [1393] Problem 6

problem number 1393

Added April 14, 2019.

Problem Chapter 6.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ c x w_x + c y w_y +(a x^2+b y^2) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   c*x*D[w[x, y,z], x] +c*y*D[w[x, y,z], y] +(a*x^2+b*y^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},-\frac {a x^2+b y^2-2 c z}{2 c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   c*x*diff(w(x,y,z),x)+c*y*diff(w(x,y,z),y)+(a*x^2+b*y^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, -\frac {a \,x^{2}+b \,y^{2}-2 z c}{2 c}\right )\]

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6.6.2.7 [1394] Problem 7

problem number 1394

Added April 14, 2019.

Problem Chapter 6.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ c z w_x -a(2 a x-b)y w_y +a (2 a x-b)z w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   c*z*D[w[x, y,z], x] -a*(2*a*x-b)*y*D[w[x, y,z], y] +a*(2*a*x-b)*z*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (c y z,\frac {-a^2 x^2+a b x+c z}{c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   c*z*diff(w(x,y,z),x)-a*(2*a*x-b)*diff(w(x,y,z),y)+a*(2*a*x-b)*z*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-x^{2} a^{2}+a b x +c z}{c}, \ln \left (c z \right )+y \right )\]

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6.6.2.8 [1395] Problem 8

problem number 1395

Added April 14, 2019.

Problem Chapter 6.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a c x^2 w_x -a c x y w_y -b^2 y^2 w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   a*c*x^2*D[w[x, y,z], x] -a*c*x*y*D[w[x, y,z], y] -b^2*y^2*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (x y,z-\frac {b^2 y^2}{3 a c x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=    a*c*x^2*diff(w(x,y,z),x) -a*c*x*y*diff(w(x,y,z),y)-b^2*y^2*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y x , \frac {3 a c x z -b^{2} y^{2}}{3 x a c}\right )\]

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6.6.2.9 [1396] Problem 9

problem number 1396

Added April 14, 2019.

Problem Chapter 6.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a x^2 w_x +b y^2 w_y +c z^2 w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   a*x^2*D[w[x, y,z], x] +b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=    a*x^2*diff(w(x,y,z),x) +b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a x -y b}{y a x}, \frac {a x -z c}{z a x}\right )\]

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6.6.2.10 [1397] Problem 10

problem number 1397

Added April 14, 2019.

Problem Chapter 6.2.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a b x^2 w_x +c z^2 w_y +2 a b x z w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   a*b*x^2*D[w[x, y,z], x] +c*z^2*D[w[x, y,z], y] +2*a*b*x*z*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {z}{x^2},y-\frac {c z^2}{3 a b x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=    a*b*x^2*diff(w(x,y,z),x) +c*z^2*diff(w(x,y,z),y)+2*a*b*x*z*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {z}{x^{2}}, \frac {3 y a b x -c \,z^{2}}{3 x a b}\right )\]

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6.6.2.11 [1398] Problem 11

problem number 1398

Added April 14, 2019.

Problem Chapter 6.2.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b c x y w_x +a^2 c x^2 w_y - b y (2 a x+c z)w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   b*c*x*y*D[w[x, y,z], x] +a^2*c*x^2*D[w[x, y,z], y] -b*y*(2*a*x+c*z)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b y^2-a^2 x^2}{2 b},\frac {x (a x+c z)}{c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=     b*c*x*y*diff(w(x,y,z),x) +a^2*c*x^2*diff(w(x,y,z),y)-b*y*(2*a*x+c*z)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a^{2} x^{2}+y^{2} b}{b}, \frac {\left (a x +c z \right ) x}{c}\right )\]

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6.6.2.12 [1399] Problem 12

problem number 1399

Added April 14, 2019.

Problem Chapter 6.2.2.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b c x y w_x +c^2 y z w_y + b^2 y^2 w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   b*c*x*y*D[w[x, y,z], x] +c^2*y*z*D[w[x, y,z], y] +b^2*y^2*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b y+c z}{2 b x},\frac {x (b y-c z)}{2 b}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := b*c*x*y*diff(w(x,y,z),x) +c^2*y*z*diff(w(x,y,z),y)+b^2*y^2*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-b^{2} y^{2}+z^{2} c^{2}}{c^{2}}, x \left (\operatorname {csgn}\left (b \right ) b y +z c \right )^{-\operatorname {csgn}\left (b \right )}\right )\]

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6.6.2.13 [1400] Problem 13

problem number 1400

Added April 14, 2019.

Problem Chapter 6.2.2.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ x y w_x +y(y-a)w_y +z(y-a) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   x*y*D[w[x, y,z], x] +y*(y-a)*D[w[x, y,z], y] +z*(y-a)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y-a}{x},\frac {z}{y}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*y*diff(w(x,y,z),x) +y*(y-a)*diff(w(x,y,z),y)+z*(y-a)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y -a}{x}, \frac {z}{y}\right )\]

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6.6.2.14 [1401] Problem 14

problem number 1401

Added April 14, 2019.

Problem Chapter 6.2.2.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b y^2 w_x -a x y w_y +c x z w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   b*y^2*D[w[x, y,z], x] -a*x*y*D[w[x, y,z], y] +c*x*z*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a x^2+b y^2}{2 b},z \left (-b y^2\right )^{\frac {c}{2 a}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := b*y^2*diff(w(x,y,z),x) -a*x*y*diff(w(x,y,z),y)+c*x*z*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {x^{2} a +b \,y^{2}}{b}, z \left (-b \,y^{2}\right )^{\frac {c}{2 a}}\right )\]

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6.6.2.15 [1402] Problem 15

problem number 1402

Added April 14, 2019.

Problem Chapter 6.2.2.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ c x z w_x + 2 a x y w_y -(2 a x+c z) z w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   c*x*z*D[w[x, y,z], x] +2*a*x*y*D[w[x, y,z], y] -(2*a*x+c*z)*z*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-c x y z,x \left (\frac {a x}{c}+z\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde := c*x*z*diff(w(x,y,z),x) +2*a*x*y*diff(w(x,y,z),y)-(2*a*x+c*z)*z*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (a x +z c \right ) x}{c}, -y c x z \right )\]

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6.6.2.16 [1403] Problem 16

problem number 1403

Added April 14, 2019.

Problem Chapter 6.2.2.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ c x z w_x + c y z w_y +a b x y w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   c*x*z*D[w[x, y,z], x] +c*y*z*D[w[x, y,z], y] +a*b*x*y*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {c z^2-a b x y}{2 c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := c*x*z*diff(w(x,y,z),x) +c*y*z*diff(w(x,y,z),y)+a*b*x*y*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, \frac {-a b x y +z^{2} c}{c}\right )\]

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6.6.2.17 [1404] Problem 17

problem number 1404

Added April 14, 2019.

Problem Chapter 6.2.2.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ c x z w_x - c y z w_y +(b y^2-a x) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   c*x*z*D[w[x, y,z], x] -c*y*z*D[w[x, y,z], y] +(b*y^2-a*x)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (x y,\frac {2 a x+b y^2+c z^2}{2 c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := c*x*z*diff(w(x,y,z),x)-c*y*z*diff(w(x,y,z),y)+(b*y^2-a*x)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y x , \frac {b \,y^{2}+c \,z^{2}+2 a x}{c}\right )\]

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6.6.2.18 [1405] Problem 18

problem number 1405

Added April 14, 2019.

Problem Chapter 6.2.2.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ c x z w_x - c y z w_y +(a x^2+b y^2 ) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   c*x*z*D[w[x, y,z], x] -c*y*z*D[w[x, y,z], y] +(a*x^2+b*y^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (x y,\frac {-a x^2+b y^2+c z^2}{2 c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := c*x*z*diff(w(x,y,z),x)-c*y*z*diff(w(x,y,z),y)+(a*x^2+b*y^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y x , \frac {-a \,x^{2}+b \,y^{2}+c \,z^{2}}{c}\right )\]

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6.6.2.19 [1406] Problem 19

problem number 1406

Added April 14, 2019.

Problem Chapter 6.2.2.19, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ x z w_x + y z w_y +(a x^2+a y^2+ b z^2 ) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   x*z*D[w[x, y,z], x] +y*z*D[w[x, y,z], y] +(a*x^2+a*y^2+b*z^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {x^{-2 b} \left (a \left (x^2+y^2\right )+(b-1) z^2\right )}{b-1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*z*diff(w(x,y,z),x)+y*z*diff(w(x,y,z),y)+(a*x^2+a*y^2+b*z^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, \frac {\left (\left (x^{2}+y^{2}\right ) a +z^{2} \left (b -1\right )\right ) x^{-2 b}}{b -1}\right )\]

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6.6.2.20 [1407] Problem 20

problem number 1407

Added April 14, 2019.

Problem Chapter 6.2.2.20, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ 2 c x z w_x + 2 c y z w_y +(c z^2-a x^2- b y^2 ) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   2*c*x*z*D[w[x, y,z], x] +2*c*y*z*D[w[x, y,z], y] +(c*z^2-a*x^2-b*y^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {a x^2+b y^2+c z^2}{c x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := 2*c*x*z*diff(w(x,y,z),x)+2*c*y*z*diff(w(x,y,z),y)+(c*z^2-a*x^2-b*y^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, \frac {a \,x^{2}+b \,y^{2}+c \,z^{2}}{c x}\right )\]

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6.6.2.21 [1408] Problem 21

problem number 1408

Added April 14, 2019.

Problem Chapter 6.2.2.21, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b c y z w_x + a c x z w_y + a b x y w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   b*c*y*z*D[w[x, y,z], x] +a*c*x*z*D[w[x, y,z], y] +a*b*x*y*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b y^2-a x^2}{2 b},\frac {c z^2-a x^2}{2 c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := b*c*y*z*diff(w(x,y,z),x)+a*c*x*z*diff(w(x,y,z),y)+a*b*x*y*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,x^{2}+y^{2} b}{b}, \frac {-a \,x^{2}+z^{2} c}{c}\right )\]

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6.6.2.22 [1409] Problem 22

problem number 1409

Added April 14, 2019.

Problem Chapter 6.2.2.22, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b c (x^2-a^2) w_x + c(b x y+a c z ) w_y + b(c x z + a b y) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   b*c*(x^2-a^2)*D[w[x, y,z], x] +c*(b*x*y+a*c*z)*D[w[x, y,z], y] +b*(c*x*z + a*b*y)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a c z+b x y}{a^2 b-b x^2},\frac {a (a b y+c x z)}{b \left (a^2-x^2\right )}\right )\right \}\right \}\]

Maple ✗

restart; 
pde :=  b*c*(x^2-a^2)*diff(w(x,y,z),x)+c*(b*x*y+a*c*z)*diff(w(x,y,z),y)+b*(c*x*z + a*b*y)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.2.23 [1410] Problem 23

problem number 1410

Added April 14, 2019.

Problem Chapter 6.2.2.23, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b x (b y +c) w_x + (b^2 y^2-a c x ) w_y + b^2 y z w_z= 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =   b*x*(b*y+c)*D[w[x, y,z], x] + (b^2*y^2-a*c*x )*D[w[x, y,z], y] + b^2*y*z*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  b*x*(b*y+c)*diff(w(x,y,z),x)+(b^2*y^2-a*c*x )*diff(w(x,y,z),y)+b^2*y*z*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (-a x -b y \right ) \ln \left (-\frac {\left (a x +b y \right ) \left (a x -c \right )}{x \left (b y +c \right )}\right )+\left (a x +b y \right ) \ln \left (\frac {-a x +c}{b y +c}\right )+\left (-a x -b y \right ) \ln \left (2\right )+b y +c}{3 a x +3 b y}, z \,{\mathrm e}^{-\frac {\int _{}^{x}\frac {2 \,{\mathrm e}^{\operatorname {RootOf}\left (-2 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (a \textit {\_a} -c \right )}{\textit {\_a}}\right ) {\mathrm e}^{\textit {\_Z}} a x -2 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (a \textit {\_a} -c \right )}{\textit {\_a}}\right ) {\mathrm e}^{\textit {\_Z}} b y +2 \,{\mathrm e}^{\textit {\_Z}} a x \ln \left (-\frac {\left (a x +b y \right ) \left (a x -c \right )}{x \left (b y +c \right )}\right )+2 \,{\mathrm e}^{\textit {\_Z}} b y \ln \left (-\frac {\left (a x +b y \right ) \left (a x -c \right )}{x \left (b y +c \right )}\right )-2 \,{\mathrm e}^{\textit {\_Z}} a x \ln \left (-\frac {a x -c}{b y +c}\right )-2 \,{\mathrm e}^{\textit {\_Z}} b y \ln \left (-\frac {a x -c}{b y +c}\right )+2 \,{\mathrm e}^{\textit {\_Z}} a x \ln \left (2\right )+2 \,{\mathrm e}^{\textit {\_Z}} b y \ln \left (2\right )+2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} a x +2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} b y +9 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (a \textit {\_a} -c \right )}{\textit {\_a}}\right ) a x +9 \ln \left (\frac {\left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) \left (a \textit {\_a} -c \right )}{\textit {\_a}}\right ) b y -2 \,{\mathrm e}^{\textit {\_Z}} b y -9 a x \ln \left (-\frac {\left (a x +b y \right ) \left (a x -c \right )}{x \left (b y +c \right )}\right )-9 b y \ln \left (-\frac {\left (a x +b y \right ) \left (a x -c \right )}{x \left (b y +c \right )}\right )+9 a x \ln \left (-\frac {a x -c}{b y +c}\right )+9 b y \ln \left (-\frac {a x -c}{b y +c}\right )-9 a x \ln \left (2\right )-9 b y \ln \left (2\right )-9 \textit {\_Z} a x -9 \textit {\_Z} b y -2 c \,{\mathrm e}^{\textit {\_Z}}-9 a x +9 c \right )} c +9 a \textit {\_a} -9 c}{\left (a \textit {\_a} -c \right ) \textit {\_a}}d \textit {\_a}}{9}}\right )\]

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6.6.2.24 [1411] Problem 24

problem number 1411

Added April 14, 2019.

Problem Chapter 6.2.2.24, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ x (b y -c z) w_x + y(c z-a x) w_y + z(a x - b y) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   x*(b*y -c*z)*D[w[x, y,z], x] + y*(c*z-a*x)*D[w[x, y,z], y] + z*(a*x - b*y)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {c x y z}{b},\frac {a x+b y+c z}{c}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   x*(b*y -c*z)*diff(w(x,y,z),x)+ y*(c*z-a*x)*diff(w(x,y,z),y)+z*(a*x - b*y)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {c_{4} c_{5} x^{c_{2}} y^{c_{2}} z^{c_{2}} {\mathrm e}^{-c_{1} x +c_{2} -\frac {c_{1} b y}{a}-\frac {c_{1} c z}{a}}}{c_{3}}\]

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6.6.2.25 [1412] Problem 25

problem number 1412

Added April 14, 2019.

Problem Chapter 6.2.2.25, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a(y+\beta )(z+\gamma ) w_x -b(x+\alpha )(z+\gamma ) w_y - c(x+\alpha )(y+\beta ) w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =   a*(y+beta)*(z+gamma)*D[w[x, y,z], x] -b*(x+alpha)*(z+gamma)*D[w[x, y,z], y] - c*(x+alpha)*(y+beta)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a y (2 \beta +y)+2 \alpha b x+b x^2}{2 a},\frac {a z (2 \gamma +z)+2 \alpha c x+c x^2}{2 a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   a*(y+beta)*(z+gamma)*diff(w(x,y,z),x)-b*(x+alpha)*(z+gamma)*diff(w(x,y,z),y)- c*(x+alpha)*(y+beta)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = c_{2} c_{4} c_{5} {\mathrm e}^{\frac {\left (y \left (y +2 \beta \right ) c_{1} +2 c_{3} x \left (\alpha +\frac {x}{2}\right )\right ) c +2 \left (a c_{3} -b c_{1} \right ) \left (\frac {z}{2}+\gamma \right ) z}{2 c}}\]

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6.6.2.26 [1413] Problem 26

problem number 1413

Added April 14, 2019.

Problem Chapter 6.2.2.26, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b c (a c x z + b^2 y^2) w_x +a c (b c y z-2 a^2 x^2)w_y - a b (2 a b x y+c^2 z^2) w_z= 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =   b*c*(a*c*x*z + b^2*y^2)*D[w[x, y,z], x] +a*c*(b*c*y*z-2*a^2*x^2)*D[w[x, y,z], y] - a*b*(2*a*b*x*y+c^2*z^2)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=   b*c*(a*c*x*z + b^2*y^2)*diff(w(x,y,z),x)+a*c*(b*c*y*z-2*a^2*x^2)*diff(w(x,y,z),y)- a*b*(2*a*b*x*y+c^2*z^2)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.6.2.27 [1414] Problem 27

problem number 1414

Added April 14, 2019.

Problem Chapter 6.2.2.27, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a(y^2+z^2) w_x +x(b z-a y)w_y -x(b y + a z) w_z= 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =   a*(y^2+z^2)*D[w[x, y,z], x] +x*(b*z-a*y)*D[w[x, y,z], y] -x*(b*y + a*z)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  a*(y^2+z^2)*diff(w(x,y,z),x)+x*(b*z-a*y)*diff(w(x,y,z),y)-x*(b*y + a*z)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = {\mathrm e}^{-\left (a \int _{}^{y}-\frac {\textit {\_a}}{-\tan \left (\operatorname {RootOf}\left (-b \ln \left (\frac {\textit {\_a}^{2}}{\cos \left (2 \textit {\_Z} \right )+1}\right )+b \ln \left (y^{2}+z^{2}\right )-2 a \arctan \left (\frac {z}{y}\right )-b \ln \left (2\right )+2 \textit {\_Z} a \right )\right ) b +a}d \textit {\_a} +a \int _{}^{y}-\frac {\textit {\_a} {\tan \left (\operatorname {RootOf}\left (-b \ln \left (\frac {\textit {\_a}^{2}}{\cos \left (2 \textit {\_Z} \right )+1}\right )+b \ln \left (y^{2}+z^{2}\right )-2 a \arctan \left (\frac {z}{y}\right )-b \ln \left (2\right )+2 \textit {\_Z} a \right )\right )}^{2}}{-\tan \left (\operatorname {RootOf}\left (-b \ln \left (\frac {\textit {\_a}^{2}}{\cos \left (2 \textit {\_Z} \right )+1}\right )+b \ln \left (y^{2}+z^{2}\right )-2 a \arctan \left (\frac {z}{y}\right )-b \ln \left (2\right )+2 \textit {\_Z} a \right )\right ) b +a}d \textit {\_a} -\frac {x^{2}}{2}\right ) \textit {\_c}_{1}} f_{5} \left (-\frac {b \ln \left (y^{2}+z^{2}\right )-2 a \arctan \left (\frac {z}{y}\right )}{2 b}\right ) c_{1}\]

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6.6.2.28 [1415] Problem 28

problem number 1415

Added April 14, 2019.

Problem Chapter 6.2.2.28, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ b(b y + c z)^2 w_x - a x(b y + 2 c z)w_y +a b x z w_z= 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  b*(b*y + c*z)^2*D[w[x, y,z], x] - a*x*(b*y + 2*c*z)*D[w[x, y,z], y] +a*b*x*z*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {z (b y+c z)}{b},-\frac {\frac {a x^2}{(b y+c z)^2}+1}{2 z^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  b*(b*y + c*z)^2*diff(w(x,y,z),x)- a*x*(b*y + 2*c*z)*diff(w(x,y,z),y)+a*b*x*z*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left (x , y , z\right ) = c_{1} {\mathrm e}^{\frac {\textit {\_c}_{1} \left (b^{2} y^{2}+b y c z +x^{2} a \right )}{2 a}} f_{5} \left (\frac {z \left (b y +c z \right )}{b}\right )\]

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6.6.2.29 [1416] Problem 29

problem number 1416

Added April 14, 2019.

Problem Chapter 6.2.2.29, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ (f_0 x - f_1) w_x + (f_0 y-f_2) w_y + (f_0 z -f_3) w_z= 0 \]

Where

\[ f_n = a_n + b_n x + c_n y+ d_n z \]

Mathematica ✗

ClearAll["Global`*"]; 
f[n_]:= a[n] + b[n]*x + c[n]*y+ d[n]*z; 
pde = (f[0]*x - f[1])*D[w[x, y,z], x] +(f[0]*y-f[2])*D[w[x, y,z], y] +(f[0]*z -f[3])*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✗

restart; 
f:= n ->  a[n] + b[n]*x + c[n]*y+ d[n]*z; 
pde :=  (f(0)*x - f(1))*diff(w(x,y,z),x)+(f(0)*y-f(2))*diff(w(x,y,z),y)+(f(0)*z -f(3))*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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