6.8.8 4.2

6.8.8.1 [1810] Problem 1

problem number 1810

Added Oct 10, 2019.

Problem Chapter 8.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \cosh ^n(\beta x) w \]

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \sqrt {-\sinh ^2(\beta x)} \text {csch}(\beta x) \cosh ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cosh ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \left (\cosh ^{n}\left (\beta x \right )\right )d x}\]

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6.8.8.2 [1811] Problem 2

problem number 1811

Added Oct 10, 2019.

Problem Chapter 8.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \cosh (\lambda x) w_z = \left ( k \cosh (\beta x)+s \cosh (\gamma z) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \lambda z -c \sinh \left (\lambda x \right )}{a \lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {k \cosh \left (\mathit {\_a} \beta \right )+s \cosh \left (\frac {\left (a \lambda z +c \sinh \left (\mathit {\_a} \lambda \right )-c \sinh \left (\lambda x \right )\right ) \gamma }{a \lambda }\right )}{a}d\mathit {\_a}}\]

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6.8.8.3 [1812] Problem 3

problem number 1812

Added Oct 10, 2019.

Problem Chapter 8.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {-\sinh ^2(\gamma x)} \text {csch}(\gamma x) \cosh ^{m+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cosh ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (\frac {a \sinh (\beta x) \cosh ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cosh ^2(\beta x)\right )}{(\beta n+\beta ) \sqrt {-\sinh ^2(\beta x)}}+y,\frac {b \sinh (\lambda x) \cosh ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cosh ^2(\lambda x)\right )}{(k \lambda +\lambda ) \sqrt {-\sinh ^2(\lambda x)}}+z\right )\right \}\right \}\]

Maple ✓

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

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

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6.8.8.4 [1813] Problem 4

problem number 1813

Added Oct 10, 2019.

Problem Chapter 8.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b \cosh (\beta y) w_y + c \cosh (\lambda x) w_z = k \cosh (\gamma z) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},z-\frac {c \sinh (\lambda x)}{a \lambda }\right ) \exp \left (\int _1^x\frac {k \cosh \left (\frac {\gamma (a \lambda z-c \sinh (\lambda x)+c \sinh (\lambda K[1]))}{a \lambda }\right )}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-b \beta x +2 a \arctan \left ({\mathrm e}^{\beta y}\right )}{b \beta }, \frac {a \lambda z -c \sinh \left (\lambda x \right )}{a \lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {k \cosh \left (\frac {\left (a \lambda z +c \sinh \left (\mathit {\_a} \lambda \right )-c \sinh \left (\lambda x \right )\right ) \gamma }{a \lambda }\right )}{a}d\mathit {\_a}}\]

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6.8.8.5 [1814] Problem 5

problem number 1814

Added Oct 10, 2019.

Problem Chapter 8.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

Maple ✓

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

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

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