6.7.16 6.3

6.7.16.1 [1685] Problem 1
6.7.16.2 [1686] Problem 2
6.7.16.3 [1687] Problem 3
6.7.16.4 [1688] Problem 4
6.7.16.5 [1689] Problem 5

6.7.16.1 [1685] Problem 1

problem number 1685

Added June 26, 2019.

Problem Chapter 7.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)+\frac {c \tan ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]

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

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

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6.7.16.2 [1686] Problem 2

problem number 1686

Added June 26, 2019.

Problem Chapter 7.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \tan (\beta z) w_z = k \tan (\lambda x)+ s \tan (\gamma y) \]

Mathematica

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

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log (\sin (\beta z))}{\beta }-\frac {c x}{a}\right )-\frac {k \log (\cos (\lambda x))}{a \lambda }-\frac {s \log (\cos (\gamma y))}{b \gamma }\right \}\right \}\]

Maple

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

\[w \left (x , y , z\right ) = \frac {2 a b \gamma \lambda \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {-\beta c x +a \ln \left (\frac {\tan \left (\beta z \right )}{\sqrt {\tan ^{2}\left (\beta z \right )+1}}\right )}{\beta c}\right )+a \lambda s \ln \left (\tan ^{2}\left (\gamma y \right )+1\right )+b \gamma k \ln \left (\tan ^{2}\left (\lambda x \right )+1\right )}{2 a b \gamma \lambda }\]

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6.7.16.3 [1687] Problem 3

problem number 1687

Added June 26, 2019.

Problem Chapter 7.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

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

Maple

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

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

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6.7.16.4 [1688] Problem 4

problem number 1688

Added June 26, 2019.

Problem Chapter 7.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \tan ^k\left (\frac {\gamma \left (-a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda x)\right ) \tan ^{n+1}(\lambda x)+a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda K[1])\right ) \tan ^{n+1}(\lambda K[1])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )+s \tan ^r\left (\frac {\mu \left (-b \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\beta x)\right ) \tan ^{m+1}(\beta x)+b \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\beta K[1])\right ) \tan ^{m+1}(\beta K[1])+\beta (m+1) z\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (z-\frac {b \tan ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\tan ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \tan ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]

Maple

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ a*tan(lambda*x)^n*diff(w(x,y,z),y)+ b*tan(beta*x)^m*diff(w(x,y,z),z)= c*tan(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'));
 

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

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6.7.16.5 [1689] Problem 5

problem number 1689

Added June 26, 2019.

Problem Chapter 7.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a_1 \tan ^{n_1}(\lambda _1 x) w_x + b_1 \tan ^{m_1}(\beta _1 y) w_y + c_1 \tan ^{k_1}(\gamma _1 z) w_z = a_2 \tan ^{n_2}(\lambda _2 x) + b_2 \tan ^{m_2}(\beta _2 y)+ c_2 \tan ^{k_2}(\gamma _2 z) \]

Mathematica

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

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

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