6.8.8 4.2

6.8.8.1 [1811] Problem 1

problem number 1811

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \! \left ( \cosh \left ( \beta \,x \right ) \right ) ^{n}c\,{\rm d}x}}\]

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6.8.8.2 [1812] Problem 2

problem number 1812

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za\lambda -c\sinh \left ( \lambda \,x \right ) }{a\lambda }} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( k\cosh \left ( \beta \,{\it \_a} \right ) +s\cosh \left ( 1/8\,{\frac {za\lambda -c\sinh \left ( \lambda \,x \right ) +c\sinh \left ( {\it \_a}\,\lambda \right ) }{a\lambda }} \right ) \right ) }{d{\it \_a}}}}\]

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6.8.8.3 [1813] Problem 3

problem number 1813

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'));
 

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

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6.8.8.4 [1814] Problem 4

problem number 1814

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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-bx\beta +2\,\arctan \left ( {{\rm e}^{\beta \,y}} \right ) a}{b\beta }},{\frac {za\lambda -c\sinh \left ( \lambda \,x \right ) }{a\lambda }} \right ) {{\rm e}^{\int ^{x}\!{\frac {k}{a}\cosh \left ( 1/8\,{\frac {za\lambda -c\sinh \left ( \lambda \,x \right ) +c\sinh \left ( {\it \_a}\,\lambda \right ) }{a\lambda }} \right ) }{d{\it \_a}}}}\]

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6.8.8.5 [1815] Problem 5

problem number 1815

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'));
 

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

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