6.7.22 7.3

6.7.22.1 [1712] Problem 1

problem number 1712

Added June 26, 2019.

Problem Chapter 7.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \arctan ^k(\lambda x)+s \]

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[lambda*x]^k+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \arctan (\lambda K[1])^k+s\right )dK[1]+c_1(y-a x,z-b x)\right \}\right \}\]

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(lambda*x)^k+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.7.22.2 [1713] Problem 2

problem number 1713

Added June 26, 2019.

Problem Chapter 7.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde :=  a1*diff(w(x,y,z),x)+ a2*diff(w(x,y,z),y)+ a3*diff(w(x,y,z),z)= b1*arctan(lambda1*x)+b2*arctan(lambda2*y)+b3*arctan(lambda3*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {2 f_{1} \left (\frac {\operatorname {a1} y -x \operatorname {a2}}{\operatorname {a1}}, \frac {\operatorname {a1} z -x \operatorname {a3}}{\operatorname {a1}}\right ) \operatorname {a1} \lambda \operatorname {1} \lambda \operatorname {2} \operatorname {a2} \lambda \operatorname {3} \operatorname {a3} -\operatorname {b1} \ln \left (\lambda \operatorname {1}^{2} x^{2}+1\right ) \lambda \operatorname {2} \operatorname {a2} \lambda \operatorname {3} \operatorname {a3} +2 \left (-\frac {\lambda \operatorname {3} \ln \left (\lambda \operatorname {2}^{2} y^{2}+1\right ) \operatorname {a1} \operatorname {a3} \operatorname {b2}}{2}+\lambda \operatorname {2} \left (-\frac {\ln \left (\lambda \operatorname {3}^{2} z^{2}+1\right ) \operatorname {a1} \operatorname {a2} \operatorname {b3}}{2}+\left (x \arctan \left (\lambda \operatorname {1} x \right ) \operatorname {a2} \operatorname {a3} \operatorname {b1} +\operatorname {a1} \left (\arctan \left (\lambda \operatorname {2} y \right ) y \operatorname {a3} \operatorname {b2} +\arctan \left (\lambda \operatorname {3} z \right ) z \operatorname {a2} \operatorname {b3} \right )\right ) \lambda \operatorname {3} \right )\right ) \lambda \operatorname {1} }{2 \operatorname {a1} \lambda \operatorname {1} \lambda \operatorname {2} \operatorname {a2} \lambda \operatorname {3} \operatorname {a3}}\]

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6.7.22.3 [1714] Problem 3

problem number 1714

Added June 26, 2019.

Problem Chapter 7.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
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)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.7.22.4 [1715] Problem 4

problem number 1715

Added June 26, 2019.

Problem Chapter 7.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*D[w[x,y,z],z]== s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

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

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6.7.22.5 [1716] Problem 5

problem number 1716

Added June 26, 2019.

Problem Chapter 7.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

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

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

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