7.2.4 2.4

7.2.4.1 [472] problem number 1

problem number 472

Added January 2, 2019.

Problem 2.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +(a \sqrt {x} y) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-\frac {2 a \,x^{\frac {3}{2}}}{3}}\right )\]

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7.2.4.2 [473] problem number 2

problem number 473

Added January 2, 2019.

Problem 2.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(a \sqrt {x} y+ b \sqrt {y}) w_y = 0 \]

Mathematica ✓

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

\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {b \operatorname {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )}{\sqrt [3]{3} a^{2/3}}-\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\\& \left \{w(x,y)\to c_1\left (\frac {b \operatorname {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )}{\sqrt [3]{3} a^{2/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\\ \end {align*}

Maple ✓

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

\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (3 \,3^{\frac {1}{3}} b x \WhittakerM \left (\frac {1}{3}, \frac {5}{6}, \frac {a \,x^{\frac {3}{2}}}{3}\right ) {\mathrm e}^{\frac {a \,x^{\frac {3}{2}}}{6}}+5 \left (a \,x^{\frac {3}{2}}\right )^{\frac {1}{3}} b x -10 \left (a \,x^{\frac {3}{2}}\right )^{\frac {1}{3}} \sqrt {y}\right ) {\mathrm e}^{-\frac {a \,x^{\frac {3}{2}}}{3}}}{10 \left (a \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}}\right )\]

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7.2.4.3 [474] problem number 3

problem number 474

Added January 2, 2019.

Problem 2.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +(a \sqrt {x} y+ b x \sqrt {y}) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt [3]{3} b \operatorname {Gamma}\left (\frac {4}{3},\frac {1}{3} a x^{3/2}\right )}{a^{4/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (3 \,3^{\frac {1}{6}} b \sqrt {x}\, \WhittakerM \left (\frac {1}{6}, \frac {2}{3}, \frac {a \,x^{\frac {3}{2}}}{3}\right ) {\mathrm e}^{\frac {a \,x^{\frac {3}{2}}}{6}}-4 \left (a \,x^{\frac {3}{2}}\right )^{\frac {1}{6}} a \sqrt {y}\right ) {\mathrm e}^{-\frac {a \,x^{\frac {3}{2}}}{3}}}{4 \left (a \,x^{\frac {3}{2}}\right )^{\frac {1}{6}} a}\right )\]

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7.2.4.4 [475] problem number 4

problem number 475

Added January 2, 2019.

Problem 2.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +A \sqrt {a x + b y+ c} w_y = 0 \]

Mathematica ✓

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

\begin {align*} & \left \{w(x,y)\to c_1\left (x-\frac {a \log \left (\frac {e^{\frac {2 \sqrt {A^2 b^2 (a x+b y+c)}}{a}}}{\left (\sqrt {A^2 b^2 (a x+b y+c)}+a\right )^2}\right )}{A^2 b^2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (x-\frac {a \log \left (\frac {e^{-\frac {2 \sqrt {A^2 b^2 (a x+b y+c)}}{a}}}{\left (a-\sqrt {A^2 b^2 (a x+b y+c)}\right )^2}\right )}{A^2 b^2}\right )\right \}\\ \end {align*}

Maple ✓

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

\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {A^{2} b^{2} x -2 \sqrt {a x +b y +c}\, A b -a \ln \left (\sqrt {a x +b y +c}\, A b -a \right )+a \ln \left (\sqrt {a x +b y +c}\, A b +a \right )+a \ln \left (\left (a x +b y +c \right ) A^{2} b^{2}-a^{2}\right )}{A^{2} b^{2}}\right )\]

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7.2.4.5 [476] problem number 5

problem number 476

Added January 2, 2019.

Problem 2.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ( a y + b \sqrt {y^2+c x^2} \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✗

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

time expired

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7.2.4.6 [477] problem number 6

problem number 477

Added January 2, 2019.

Problem 2.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left (a x + b \sqrt {y} \right ) w_x - \left ( c \sqrt {x} + a y \right ) w_y = 0 \]

Mathematica ✓

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

\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {3 a^3 x^3}{8 b^2}+\frac {2}{3} c x^{3/2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (a x y-\frac {2}{3} b y^{3/2}+\frac {2}{3} c x^{3/2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (a x y+\frac {2}{3} b y^{3/2}+\frac {2}{3} c x^{3/2}\right )\right \}\\ \end {align*}

Maple ✓

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

\[w \left (x , y\right ) = \textit {\_F1} \left (\RootOf \left (3 a^{4} y^{4}+8 a b \,c^{2} y^{\frac {5}{2}}-2 \left (-a^{3} y^{3}-4 b \,c^{2} y^{\frac {3}{2}}-6 \textit {\_Z} \,c^{2}+2 \sqrt {2 a^{3} b \,y^{\frac {9}{2}}+3 \textit {\_Z} \,a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \textit {\_Z} b \,c^{2} y^{\frac {3}{2}}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{\frac {1}{3}} a^{3} y^{3}+12 \textit {\_Z} a \,c^{2} y +3 \left (-a^{3} y^{3}-4 b \,c^{2} y^{\frac {3}{2}}-6 \textit {\_Z} \,c^{2}+2 \sqrt {2 a^{3} b \,y^{\frac {9}{2}}+3 \textit {\_Z} \,a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \textit {\_Z} b \,c^{2} y^{\frac {3}{2}}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{\frac {2}{3}} a^{2} y^{2}-4 \sqrt {2 a^{3} b \,y^{\frac {9}{2}}+3 \textit {\_Z} \,a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \textit {\_Z} b \,c^{2} y^{\frac {3}{2}}+9 \textit {\_Z}^{2} c^{2}}\, a c y -4 \left (-a^{3} y^{3}-4 b \,c^{2} y^{\frac {3}{2}}-6 \textit {\_Z} \,c^{2}+2 \sqrt {2 a^{3} b \,y^{\frac {9}{2}}+3 \textit {\_Z} \,a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \textit {\_Z} b \,c^{2} y^{\frac {3}{2}}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{\frac {2}{3}} c^{2} x +\left (-a^{3} y^{3}-4 b \,c^{2} y^{\frac {3}{2}}-6 \textit {\_Z} \,c^{2}+2 \sqrt {2 a^{3} b \,y^{\frac {9}{2}}+3 \textit {\_Z} \,a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \textit {\_Z} b \,c^{2} y^{\frac {3}{2}}+9 \textit {\_Z}^{2} c^{2}}\, c \right )^{\frac {4}{3}}\right )\right )\]

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7.2.4.7 [478] problem number 7

problem number 478

Added January 2, 2019.

Problem 2.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \sqrt {f(x)} w_x + \sqrt {f(y)} w_y = 0 \] Where \(f(t) = \sum _{n=0}^{4} a_n t^n \)

Mathematica ✗

ClearAll["Global`*"]; 
f[t_] := Sum[a[n]*t^n, {n, 1, 4}]; 
pde =  Sqrt[f[x]]*D[w[x, y], x] + Sqrt[f[y]]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
f:=t->sum(a[n]*t^n,n=1..4); 
pde := sqrt(f(x))* diff(w(x,y),x)+ sqrt(f(y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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