6.7.19 6.5

6.7.19.1 [1696] Problem 1

problem number 1696

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\beta x)\right )}{\beta m+\beta }+z,y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )+\frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\gamma x)\right )}{\gamma k+\gamma }+s x\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = s x +\int c \left (\sin ^{k}\left (\gamma x \right )\right )d x +\mathit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int b \left (\cos ^{m}\left (\beta x \right )\right )d x \right )\right )\]

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6.7.19.2 [1697] Problem 2

problem number 1697

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'));
 

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

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6.7.19.3 [1698] Problem 3

problem number 1698

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}\left (c \left (\cos ^{k}\left (\left (a \left (\int \left (\cos ^{n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )+y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \gamma \right )\right )+s \left (\frac {\sin \left (\left (z +\int b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )\right )-1}\right )^{m}d \mathit {\_f} -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )-1}\right )^{m}d\mathit {\_b} \right )\right ) \mu \right )}{\cos \left (\left (z +\int b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )\right )-1}\right )^{m}d \mathit {\_f} -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )-1}\right )^{m}d\mathit {\_b} \right )\right ) \mu \right )}\right )^{r}\right )d\mathit {\_f} +\mathit {\_F1} \left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cos ^{n}\left (\mathit {\_b} \lambda \right )\right )d \mathit {\_b} \right )\right )-1}\right )^{m}d\mathit {\_b} \right )\right )\]

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6.7.19.4 [1699] Problem 4

problem number 1699

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'));
 

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

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6.7.19.5 [1700] Problem 5

problem number 1700

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

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