Added Oct 18, 2019.
Problem Chapter 8.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \sin ^n(\lambda x) w \]
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*Sin[lambda*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
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*sin(lambda*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \sin (\lambda z) w_z = \left ( k \sin (\gamma x)+s \sin (\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Sin[lambda*z]*D[w[x,y,z],z]== (k*Sin[gamma*x]+s*Sin[beta*y])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{-\frac {k \cos (\gamma x)}{a \gamma }-\frac {s \cos (\beta y)}{b \beta }} c_1\left (y-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\lambda z}{2}\right )\right )}{\lambda }-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*sin(lambda*z)*diff(w(x,y,z),z)= (k*sin(gamma*x)+s*sin(beta*y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {a}{c\lambda }\ln \left ( \RootOf \left ( \lambda \,z-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {xc\lambda }{a}}}} \left ( {{\rm e}^{2\,{\frac {xc\lambda }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}},-{ \left ( {{\rm e}^{2\,{\frac {xc\lambda }{a}}}}{{\it \_Z}}^{2}-1 \right ) \left ( {{\rm e}^{2\,{\frac {xc\lambda }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) {{\rm e}^{{\frac {-a\gamma \,s\cos \left ( \beta \,y \right ) -k\cos \left ( \gamma \,x \right ) \beta \,b}{a\gamma \,\beta \,b}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta x) w_z = c \sin ^k(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*z]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*z)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -y+\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x,-\int ^{y}\! \left ( \sin \left ( \lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!a \left ( \sin \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}{d{\it \_b}}-y+\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x \right ) \right ) \right ) ^{-n}{d{\it \_b}}+\int \!{\frac { \left ( \sin \left ( \beta \,z \right ) \right ) ^{-m}a}{b}}\,{\rm d}z \right ) {{\rm e}^{\int ^{y}\!{\frac {c \left ( \sin \left ( \gamma \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!a \left ( \sin \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}{d{\it \_a}}-y+\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x \right ) \right ) \right ) ^{k} \left ( \sin \left ( \lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!a \left ( \sin \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}{d{\it \_a}}-y+\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x \right ) \right ) \right ) ^{-n}}{a}}{d{\it \_b}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta y) w_z = \left ( c \sin ^k(\gamma y) + s \sin ^r(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*y]^m*D[w[x,y,z],z]== (c*Sin[gamma*y]^k+s*Sin[mu*z]^r)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*y)^m*diff(w(x,y,z),z)= (c*sin(gamma*y)^k+s*sin(mu*z)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( \int \! \left ( \sin \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) {{\rm e}^{\int ^{x}\!c \left ( -\sin \left ( \gamma \, \left ( -\int \!a \left ( \sin \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}+\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x-y \right ) \right ) \right ) ^{k}+ \left ( \sin \left ( \mu \, \left ( \int \!b \left ( \sin \left ( \beta \, \left ( a\int \! \left ( \sin \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}-\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_g}-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( \int \! \left ( \sin \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{r}s{d{\it \_g}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \sin (\beta y) w_y + c \sin (\lambda x) w_z = k \sin (\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] + c*Sin[lambda*x]^m*D[w[x,y,z],z]== k*Sin[gamma*z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},z-\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{m+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\lambda x)\right )}{a \lambda m+a \lambda }\right ) \exp \left (\int _1^x\frac {k \sin \left (\frac {\gamma \left (-c \sqrt {\cos ^2(\lambda x)} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{m+1}(\lambda x)+c \sqrt {\cos ^2(\lambda K[1])} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\lambda K[1])\right ) \sec (\lambda K[1]) \sin ^{m+1}(\lambda K[1])+a \lambda (m+1) z\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*sin(beta*y)*diff(w(x,y,z),y)+ c*sin(lambda*x)^m*diff(w(x,y,z),z)= k*sin(gamma*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{b\beta }\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {xb\beta }{a}}}} \left ( {{\rm e}^{2\,{\frac {xb\beta }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}},-{ \left ( {{\rm e}^{2\,{\frac {xb\beta }{a}}}}{{\it \_Z}}^{2}-1 \right ) \left ( {{\rm e}^{2\,{\frac {xb\beta }{a}}}}{{\it \_Z}}^{2}+1 \right ) ^{-1}} \right ) \right ) \right ) },-\int \!{\frac {c \left ( \sin \left ( \lambda \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+z \right ) {{\rm e}^{\int ^{x}\!-{\frac {k}{a}\sin \left ( \gamma \, \left ( -\int \!{\frac {c \left ( \sin \left ( \lambda \,{\it \_b} \right ) \right ) ^{m}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac {c \left ( \sin \left ( \lambda \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-z \right ) \right ) }{d{\it \_b}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \sin ^{m_1}(\beta _1 y) w_y + c_1 \sin ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \sin ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y) + c_2 \sin ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Sin[lambda1*z]^n1*D[w[x, y,z], x] + b1*Sin[beta1*y]^m1*D[w[x, y,z], y] + c1*Sin[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Sin[lambda2*z]^n2 + b2*Sin[beta2*y]^m2 + c2*Sin[gamma2*z]^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*sin(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*sin(beta1*y)^m1*diff(w(x,y,z),y)+ c1*sin(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*sin(lambda2*z)^n2 + b2*sin(beta2*y)^m2 + c2*sin(gamma2*z)^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 ( \sin \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \sin \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z,-\int ^{y}\!{\frac {{\it a1}\, \left ( \sin \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( \sin \left ( \lambda 1\,\RootOf \left ( \int \! \left ( \sin \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \sin \left ( \gamma 1\,{\it \_b} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_b}}-\int \! \left ( \sin \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \sin \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n1}}}{d{\it \_f}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \sin \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( {\it a2}\, \left ( \sin \left ( \lambda 2\,\RootOf \left ( \int \! \left ( \sin \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \sin \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \sin \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \sin \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n2}}+{\it c2}\, \left ( \sin \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \sin \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \sin \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \sin \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \sin \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}+{\it b2}\, \left ( \sin \left ( \beta 2\,{\it \_f} \right ) \right ) ^{{\it m2}} \right ) }{d{\it \_f}}}}\]
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