6.8.7 4.1

6.8.7.1 [1806] Problem 1

problem number 1806

Added Oct 10, 2019.

Problem Chapter 8.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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6.8.7.2 [1807] Problem 2

problem number 1807

Added Oct 10, 2019.

Problem Chapter 8.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.8.7.3 [1808] Problem 3

problem number 1808

Added Oct 10, 2019.

Problem Chapter 8.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

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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6.8.7.4 [1809] Problem 4

problem number 1809

Added Oct 10, 2019.

Problem Chapter 8.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.8.7.5 [1810] Problem 5

problem number 1810

Added Oct 10, 2019.

Problem Chapter 8.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

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

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