6.7.19 6.5

6.7.19.1 [1697] Problem 1

problem number 1697

Added June 26, 2019.

Problem Chapter 7.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\beta x)\right )}{\beta m+\beta }+z,y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )+\frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\gamma x)\right )}{\gamma k+\gamma }+s x\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y,z \right ) =\int \!c \left ( \sin \left ( x/8 \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cos \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \]

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6.7.19.2 [1698] Problem 2

problem number 1698

Added June 26, 2019.

Problem Chapter 7.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( \cos \left ( 1/8\,a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f}-1/8\,\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y/8 \right ) \right ) ^{k}+s \left ( \sin \left ( \mu \, \left ( b\int \! \left ( \sin \left ( \beta \, \left ( a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_f}-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( \int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{r}{d{\it \_f}}+{\it \_F1} \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( \int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \]

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6.7.19.3 [1699] Problem 3

problem number 1699

Added June 26, 2019.

Problem Chapter 7.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( \cos \left ( 1/8\,a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f}-1/8\,\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y/8 \right ) \right ) ^{k}+s \left ( \sin \left ( \mu \, \left ( \int \!b \left ( {\frac {-\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) }{\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) -1}} \right ) ^{m}\,{\rm d}{\it \_f}-\int ^{x}\!b \left ( {\frac {-\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \left ( \cos \left ( \mu \, \left ( \int \!b \left ( {\frac {-\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) }{\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f} \right ) -1}} \right ) ^{m}\,{\rm d}{\it \_f}-\int ^{x}\!b \left ( {\frac {-\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{-1} \right ) ^{r}{d{\it \_f}}+{\it \_F1} \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( {\frac {-\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \beta \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{m}{d{\it \_b}}+z \right ) \]

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6.7.19.4 [1700] Problem 4

problem number 1700

Added June 26, 2019.

Problem Chapter 7.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a1 \sin ^{n1}(\lambda _1 x) w_x + b1 \cos ^{m1}(\beta _1 y) w_y + c1 \cos ^{k1}(\gamma _1 z) w_z = a2 \cos ^{n2}(\lambda _2 x) + b2 \sin ^{m2}(\beta _2 y) + c2 \cos ^{k2}(\gamma _2 z) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Sin[lambda1*x]^n1*D[w[x, y,z], x] + b1*Cos[beta1*y]^m1*D[w[x, y,z], y] + c1*Cos[gamma1*z]^k1*D[w[x,y,z],z]==a2*Cos[lambda2*x]^n2 + b2*Sin[beta2*y]^m2 + c2*Cos[gamma2*z]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$Aborted

Maple ✓

restart; 
local gamma; 
pde :=  a1*sin(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*cos(beta1*x)^m1*diff(w(x,y,z),y)+ c1*cos(gamma1*z)^k1*diff(w(x,y,z),z)= a2*cos(lambda2*x)^n2+ b2*sin(beta2*x)^m2+ c2*cos(gamma2*z)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac { \left ( \sin \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( \left ( \cos \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \sin \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}{\it c2}+{\it b2}\, \left ( \sin \left ( \beta 2\,{\it \_f} \right ) \right ) ^{{\it m2}}+{\it a2}\, \left ( \cos \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}} \right ) }{d{\it \_f}}+{\it \_F1} \left ( {\frac {y{\it a1}-{\it b1}\,\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}} \left ( \cos \left ( \beta 1\,x \right ) \right ) ^{{\it m1}}\,{\rm d}x}{{\it a1}}},-\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \]

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6.7.19.5 [1701] Problem 5

problem number 1701

Added June 26, 2019.

Problem Chapter 7.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a1 \tan ^{n1}(\lambda _1 x) w_x + b1 \cot ^{m1}(\beta _1 y) w_y + c1 \cot ^{k1}(\gamma _1 z) w_z = a2 \cot ^{n2}(\lambda _2 x) + b2 \tan ^{m2}(\beta _2 y) + c2 \cot ^{k2}(\gamma _2 z) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Tan[lambda1*x]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cot[gamma1*z]^k1*D[w[x,y,z],z]==a2*Cot[lambda2*x]^n2 + b2*Tan[beta2*y]^m2 + c2*Cot[gamma2*z]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a1*tan(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*x)^m1*diff(w(x,y,z),y)+ c1*cot(gamma1*z)^k1*diff(w(x,y,z),z)= a2*cot(lambda2*x)^n2+ b2*tan(beta2*x)^m2+ c2*cot(gamma2*z)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac { \left ( \tan \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( \left ( {\frac {\cos \left ( \lambda 2\,{\it \_f} \right ) }{\sin \left ( \lambda 2\,{\it \_f} \right ) }} \right ) ^{{\it n2}}{\it a2}+ \left ( \cos \left ( \gamma 2\,\RootOf \left ( -\int \! \left ( \tan \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}+\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}+\int \! \left ( {\frac {\sin \left ( \lambda 1\,x \right ) }{\cos \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x-\int \!{\frac {{\it a1}}{{\it c1}} \left ( {\frac {\cos \left ( \gamma 1\,z \right ) }{\sin \left ( \gamma 1\,z \right ) }} \right ) ^{-{\it k1}}}\,{\rm d}z \right ) \right ) \left ( \sin \left ( \gamma 2\,\RootOf \left ( -\int \! \left ( \tan \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}+\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}+\int \! \left ( {\frac {\sin \left ( \lambda 1\,x \right ) }{\cos \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x-\int \!{\frac {{\it a1}}{{\it c1}} \left ( {\frac {\cos \left ( \gamma 1\,z \right ) }{\sin \left ( \gamma 1\,z \right ) }} \right ) ^{-{\it k1}}}\,{\rm d}z \right ) \right ) \right ) ^{-1} \right ) ^{{\it k2}}{\it c2}+{\it b2}\, \left ( {\frac {\sin \left ( \beta 2\,{\it \_f} \right ) }{\cos \left ( \beta 2\,{\it \_f} \right ) }} \right ) ^{{\it m2}} \right ) }{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{{\it a1}} \left ( y{\it a1}-{\it b1}\,\int \! \left ( {\frac {\cos \left ( \beta 1\,x \right ) }{\sin \left ( \beta 1\,x \right ) }} \right ) ^{{\it m1}} \left ( {\frac {\sin \left ( \lambda 1\,x \right ) }{\cos \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x \right ) },-\int \! \left ( {\frac {\sin \left ( \lambda 1\,x \right ) }{\cos \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac {{\it a1}}{{\it c1}} \left ( {\frac {\cos \left ( \gamma 1\,z \right ) }{\sin \left ( \gamma 1\,z \right ) }} \right ) ^{-{\it k1}}}\,{\rm d}z \right ) \]

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