Added June 20, 2019.
Problem Chapter 7.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \tanh ^k(\lambda x)+s \]
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*Tanh[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*tanh(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-bx+z \right ) \]
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Added June 20, 2019.
Problem Chapter 7.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \tanh (\lambda x) w_z = k \tanh (\beta y)+s \tanh (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Tanh[lambda*x]*D[w[x,y,z],z]== k*Tanh[beta*y]+s*Tanh[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \tanh \left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )+s \tanh \left (\frac {\gamma (a \lambda z-c \log (\cosh (\lambda x))+c \log (\cosh (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a},z-\frac {c \log (\cosh (\lambda x))}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*tanh(lambda*x)*diff(w(x,y,z),z)=k*tanh(beta*y)+s*tanh(gamma*z); 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 {1}{a} \left ( \left ( -k-s \right ) \sinh \left ( 1/16\,{\frac {c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) -16\,\lambda \, \left ( \left ( \beta \,y+z/8 \right ) a-b\beta \, \left ( x-{\it \_a} \right ) \right ) }{a\lambda }} \right ) +\sinh \left ( 1/16\,{\frac {c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) +16\, \left ( \left ( \beta \,y-z/8 \right ) a-b\beta \, \left ( x-{\it \_a} \right ) \right ) \lambda }{a\lambda }} \right ) \left ( k-s \right ) \right ) \left ( \cosh \left ( 1/16\,{\frac {c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) +16\, \left ( \left ( \beta \,y-z/8 \right ) a-b\beta \, \left ( x-{\it \_a} \right ) \right ) \lambda }{a\lambda }} \right ) +\cosh \left ( 1/16\,{\frac {c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) -16\,\lambda \, \left ( \left ( \beta \,y+z/8 \right ) a-b\beta \, \left ( x-{\it \_a} \right ) \right ) }{a\lambda }} \right ) \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}},1/2\,{\frac {2\,za\lambda +c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) +c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) }{a\lambda }} \right ) \]
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Added June 19, 2019.
Problem Chapter 7.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \tanh ^n(\beta x) w_y + c \tanh ^k(\lambda x) w_z = c \tanh ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Tanh[beta*x]^n*D[w[x, y,z], y] + b*Tanh[lambda*x]^k*D[w[x,y,z],z]== c*Tanh[gamma*x]^m+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )+\frac {c \tanh ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\tanh ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*tanh(beta*x)^n*diff(w(x,y,z),y)+ b*tanh(lambda*x)^k*diff(w(x,y,z),z)=c*tanh(gamma*x)^m+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \tanh \left ( x/8 \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \tanh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]
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Added June 19, 2019.
Problem Chapter 7.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \tanh (\beta y) w_y + c \tanh (\lambda x) w_z = k \tanh (\gamma z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] + c*Tanh[lambda*x]*D[w[x,y,z],z]== k*Tanh[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+ c*tanh(lambda*x)*diff(w(x,y,z),z)=k*tanh(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{y}\!{\frac {k}{b\tanh \left ( \beta \,{\it \_a} \right ) }\sinh \left ( 1/8\,{\frac {1}{a\lambda } \left ( za\lambda -c\ln \left ( \cosh \left ( \lambda \,x \right ) \right ) +c\ln \left ( \cosh \left ( 1/2\,{\frac {\lambda \, \left ( -2\,bx\beta -a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) -a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) \right ) }{b\beta }} \right ) \right ) \right ) } \right ) \left ( \cosh \left ( 1/8\,{\frac {1}{a\lambda } \left ( za\lambda -c\ln \left ( \cosh \left ( \lambda \,x \right ) \right ) +c\ln \left ( \cosh \left ( 1/2\,{\frac {\lambda \, \left ( -2\,bx\beta -a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) -a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) \right ) }{b\beta }} \right ) \right ) \right ) } \right ) \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) }{b\beta }},{\frac {za\lambda -c\ln \left ( \cosh \left ( \lambda \,x \right ) \right ) }{a\lambda }} \right ) \]
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Added June 19, 2019.
Problem Chapter 7.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \tanh (\lambda x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] + c*Tanh[gamma*z]*D[w[x,y,z],z]== k*Tanh[lambda*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {1}{2} \left (\frac {\log (\sinh (\beta y))}{\beta }-\frac {b x}{a}\right ),\frac {b \log \left (\sinh ^2(\gamma z)\right )}{\gamma }-\frac {2 c \log (\sinh (\beta y))}{\beta }\right )+\frac {k \log (\cosh (\lambda x))}{a \lambda }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+ c*tanh(gamma*z)*diff(w(x,y,z),z)=k*tanh(lambda*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) }{b\beta }},1/2\,{\frac {1}{\beta \,c} \left ( 8\,\ln \left ( {\frac { \left ( \RootOf \left ( 8\,\arctanh \left ( {\it \_Z} \right ) +z \right ) \right ) ^{2}}{ \left ( \RootOf \left ( 8\,\arctanh \left ( {\it \_Z} \right ) +z \right ) \right ) ^{2}-1}} \right ) b\beta +2\,\beta \,cy+c\ln \left ( -4\, \left ( {{\rm e}^{2\,\beta \,y}}-1 \right ) ^{-2} \right ) \right ) } \right ) a\lambda +k\ln \left ( \cosh \left ( \lambda \,x \right ) \right ) \right ) }\]
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Added June 19, 2019.
Problem Chapter 7.4.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a \tanh (\lambda x) w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Tanh[lambda*x]*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] + c*Tanh[gamma*z]*D[w[x,y,z],z]== k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*tanh(lambda*x)*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+ c*tanh(gamma*z)*diff(w(x,y,z),z)=k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left ( x,y,z \right ) =1/2\,{\frac {1}{{\it \_C2}\,a\lambda } \left ( 2\,{{\it \_C2}}^{2}{\it \_C1}\, \left ( {\frac {\sinh \left ( \lambda \,x \right ) -\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-1/2\,{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\frac {\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-1/2\,{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) -\cosh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{-1/2\,{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) +\cosh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{-1/2\,{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{{\frac {{\it \_c}_{{2}}}{\beta }}}{\it \_C3}\, \left ( \left ( {\frac {\sinh \left ( z/8 \right ) +\cosh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{4} \left ( \left ( {\frac {\sinh \left ( z/8 \right ) +\cosh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{4} \left ( \left ( {\frac {\sinh \left ( z/8 \right ) -\cosh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{4} \left ( \left ( {\frac {\sinh \left ( z/8 \right ) -\cosh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{4}a\lambda +2\,k \left ( \left ( {\frac {\sinh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{8} \left ( \left ( {\frac {\sinh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{8} \left ( {\it \_C1}\,\lambda -\ln \left ( {{\rm e}^{\lambda \,x}} \right ) {\it \_C2}+1/2\,\ln \left ( {{\rm e}^{\lambda \,x}}-1 \right ) {\it \_C2}+1/2\,\ln \left ( {{\rm e}^{\lambda \,x}}+1 \right ) {\it \_C2}+1/2\,\ln \left ( {{\rm e}^{2\,\lambda \,x}}-1 \right ) {\it \_C2} \right ) \right ) \left ( \left ( {\frac {\sinh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{c}}} \right ) ^{-8} \left ( \left ( {\frac {\sinh \left ( z/8 \right ) }{\cosh \left ( z/8 \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{c}}} \right ) ^{-8}}\]
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Added June 19, 2019.
Problem Chapter 7.4.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \tanh ^{n_1}(\lambda _1 x) w_x + b_1 \tanh ^{m_1}(\beta _1 y) w_y + c_1 \tanh ^{k_1}(\gamma _1 z) w_z = a_2 \tanh ^{n_2}(\lambda _2 x) + b_2 \tanh ^{m_2}(\beta _2 y) w_y + c_2 \tanh ^{k_2}(\gamma _2 z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a1*Tanh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Tanh[beta1*x]^m1*D[w[x, y,z], y] + c1*Tanh[gamma1*x]^k1*D[w[x,y,z],z]== a2*Tanh[lambda1*x]^n2 + b2*Tanh[beta2*x]^m2 + c2*Tanh[gamma2*x]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^x\frac {\text {b1} \tanh ^{\text {m1}}(\text {beta1} K[1]) \tanh ^{-\text {n1}}(\text {lambda1} K[1])}{\text {a1}}dK[1],z-\int _1^x\frac {\text {c1} \tanh ^{\text {k1}}(\text {gamma1} K[2]) \tanh ^{-\text {n1}}(\text {lambda1} K[2])}{\text {a1}}dK[2]\right )+\int _1^x\frac {\tanh ^{-\text {n1}}(\text {lambda1} K[3]) \left (\text {c2} \tanh ^{\text {k2}}(\text {gamma2} K[3])+\text {b2} \tanh ^{\text {m2}}(\text {beta2} K[3])+\text {a2} \tanh ^{\text {n2}}(\text {lambda1} K[3])\right )}{\text {a1}}dK[3]\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a1*tanh(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*tanh(beta1*x)^m1*diff(w(x,y,z),y)+ c1*tanh(gamma1*x)^k1*diff(w(x,y,z),z)=a2*tanh(lambda1*x)^n2 + b2*tanh(beta2*x)^m2 + c2*tanh(gamma2*x)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!{\frac { \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}+{\it n2}}{\it a2}+ \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}{\it b2}\, \left ( \tanh \left ( \beta 2\,x \right ) \right ) ^{{\it m2}}+ \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}{\it c2}\, \left ( \tanh \left ( \gamma 2\,x \right ) \right ) ^{{\it k2}}}{{\it a1}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{{\it a1}} \left ( y{\it a1}-{\it b1}\,\int \! \left ( {\frac {\sinh \left ( \beta 1\,x \right ) }{\cosh \left ( \beta 1\,x \right ) }} \right ) ^{{\it m1}} \left ( {\frac {\sinh \left ( \lambda 1\,x \right ) }{\cosh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x \right ) },{\frac {1}{{\it a1}} \left ( z{\it a1}-{\it c1}\,\int \! \left ( {\frac {\sinh \left ( \gamma 1\,x \right ) }{\cosh \left ( \gamma 1\,x \right ) }} \right ) ^{{\it k1}} \left ( {\frac {\sinh \left ( \lambda 1\,x \right ) }{\cosh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x \right ) } \right ) \]
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