6.8.21 7.3

6.8.21.1 [1881] Problem 1

problem number 1881

Added Nov 30, 2019.

Problem Chapter 8.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \arctan ^n(\beta x) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\int _1^xc \arctan (\beta K[1])^ndK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = f_{1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{c \int \arctan \left (\beta x \right )^{n}d x}\]

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6.8.21.2 [1882] Problem 2

problem number 1882

Added Nov 30, 2019.

Problem Chapter 8.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a_1 w_x + a_2 w_y + a_3 w_z = \left ( b_1 \arctan (\lambda _1 x)+b_2 \arctan (\lambda _2 y)+b_3 \arctan (\lambda _3 z) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*D[w[x,y,z],x]+a2*D[w[x,y,z],y]+a3*D[w[x,y,z],z]== (b1*ArcTan[lambda1*x]+b2*ArcTan[lambda2*y]+b3*ArcTan[lambda3*z] ) * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \left (\text {lambda1}^2 x^2+1\right )^{-\frac {\text {b1}}{2 \text {a1} \text {lambda1}}} \left (\text {a1}^2 \left (\text {lambda2}^2 y^2+1\right )\right )^{-\frac {\text {b2}}{2 \text {a2} \text {lambda2}}} \left (\text {a1}^2 \left (\text {lambda3}^2 z^2+1\right )\right )^{-\frac {\text {b3}}{2 \text {a3} \text {lambda3}}} c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (\frac {\text {b1} x \arctan (\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} y \arctan (\text {lambda2} y)}{\text {a2}}+\frac {\text {b3} z \arctan (\text {lambda3} z)}{\text {a3}}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  a__1*diff(w(x,y,z),x)+ a__2*diff(w(x,y,z),y)+ a__3*diff(w(x,y,z),z)= (b__1*arctan(lambda__1*x)+b__2*arctan(lambda__2*y)+b__3*arctan(lambda__3*z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a_{1} y -x a_{2}}{a_{1}}, \frac {a_{1} z -a_{3} x}{a_{1}}\right ) \left (\lambda _{1}^{2} x^{2}+1\right )^{-\frac {b_{1}}{2 a_{1} \lambda _{1}}} \left (\lambda _{2}^{2} y^{2}+1\right )^{-\frac {b_{2}}{2 \lambda _{2} a_{2}}} \left (\lambda _{3}^{2} z^{2}+1\right )^{-\frac {b_{3}}{2 \lambda _{3} a_{3}}} {\mathrm e}^{\frac {\arctan \left (\lambda _{3} z \right ) a_{1} a_{2} b_{3} z +\arctan \left (\lambda _{2} y \right ) a_{1} a_{3} b_{2} y +b_{1} x \arctan \left (\lambda _{1} x \right ) a_{2} a_{3}}{a_{1} a_{2} a_{3}}}\]

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6.8.21.3 [1883] Problem 3

problem number 1883

Added Nov 30, 2019.

Problem Chapter 8.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \arctan ^n(\lambda x) \arctan ^k(\beta z) w_z = s \arctan ^m(\gamma x) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcTan[lambda*x]^n*ArcTan[beta*z]^k*D[w[x,y,z],z]==s*ArcTan[gamma*x]^m * w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*arctan(lambda*x)^n*arctan(beta*z)^k*diff(w(x,y,z),z)= s*arctan(gamma*x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, -\int \arctan \left (\lambda x \right )^{n}d x +\frac {a \int \arctan \left (\beta z \right )^{-k}d z}{c}\right ) {\mathrm e}^{\frac {s x \arctan \left (\gamma x \right )}{a}} \left (\gamma ^{2} x^{2}+1\right )^{-\frac {s}{2 a \gamma }}\]

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6.8.21.4 [1884] Problem 4

problem number 1884

Added Nov 30, 2019.

Problem Chapter 8.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \arctan ^n(\lambda x) \arctan ^m(\beta y) \arctan ^k(\gamma z) w_z = s w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcTan[lambda*x]^n*ArcTan[beta*y]^m*ArcTan[gamma*z]^k*D[w[x,y,z],z]==s* w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*arctan(lambda*x)^n*arctan(beta*y)^m*arctan(gamma*z)^k*diff(w(x,y,z),z)= s*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, -\int _{}^{x}\arctan \left (\lambda \textit {\_a} \right )^{n} \arctan \left (\frac {\left (a y -b \left (-\textit {\_a} +x \right )\right ) \beta }{a}\right )^{m}d \textit {\_a} +\frac {a \int \arctan \left (\gamma z \right )^{-k}d z}{c}\right ) {\mathrm e}^{\frac {s x}{a}}\]

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6.8.21.5 [1885] Problem 5

problem number 1885

Added Nov 30, 2019.

Problem Chapter 8.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arctan ^n(\lambda x) w_y + c \arctan ^k(\beta z) w_z = s \arctan ^m(\gamma x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x,y,z],x]+b*ArcTan[lambda*x]^n*D[w[x,y,z],y]+c*ArcTan[beta*z]^k*D[w[x,y,z],z]==s* ArcTan[gamma*x]^m*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^z\frac {s \arctan (\beta K[3])^{-k} \arctan \left (\frac {\gamma \left (c x-a \int _1^z\arctan (\beta K[2])^{-k}dK[2]+a \int _1^{K[3]}\arctan (\beta K[2])^{-k}dK[2]\right )}{c}\right ){}^m}{c}dK[3]\right ) c_1\left (y-\int _1^x\frac {b \arctan (\lambda K[1])^n}{a}dK[1],\int _1^z\arctan (\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  a*diff(w(x,y,z),x)+ b*arctan(lambda*x)^n*diff(w(x,y,z),y)+ c*arctan(beta*z)^k*diff(w(x,y,z),z)= s*arctan(gamma*x)^m*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-y +\frac {b \int \arctan \left (\lambda x \right )^{n}d x}{a}, -\int _{}^{y}{\arctan \left (\lambda \operatorname {RootOf}\left (-b \int _{}^{\textit {\_Z}}\arctan \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} +b \int \arctan \left (\lambda x \right )^{n}d x +\textit {\_b} a -y a \right )\right )}^{-n}d \textit {\_b} +\frac {b \int \arctan \left (\beta z \right )^{-k}d z}{c}\right ) {\mathrm e}^{\frac {s \int _{}^{y}{\arctan \left (\gamma \operatorname {RootOf}\left (-b \int _{}^{\textit {\_Z}}\arctan \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} +b \int \arctan \left (\lambda x \right )^{n}d x +\textit {\_b} a -y a \right )\right )}^{m} {\arctan \left (\lambda \operatorname {RootOf}\left (-b \int _{}^{\textit {\_Z}}\arctan \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} +b \int \arctan \left (\lambda x \right )^{n}d x +\textit {\_b} a -y a \right )\right )}^{-n}d \textit {\_b}}{b}}\]

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