Added June 26, 2019.
Problem Chapter 7.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \sin ^k(\lambda x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + c*D[w[x,y,z],z]== c*Sin[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)+\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*sin(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \sin \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \sin (\gamma z) w_z = k \sin (\alpha x)+ s \sin (\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Sin[gamma*z]*D[w[x,y,z],z]== k*Sin[alpha*x]+s*Sin[beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )-\frac {k \cos (\alpha x)}{a \alpha }-\frac {s \cos (\beta y)}{b \beta }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*sin(gamma*z)*diff(w(x,y,z),z)= k*sin(alpha*x)+s*sin(beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{\beta \,ba\alpha } \left ( {\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {a}{c\gamma }\ln \left ( \RootOf \left ( \gamma \,z-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {c\gamma \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \beta \,ba\alpha -\cos \left ( \alpha \,x \right ) k\beta \,b-sa\cos \left ( \beta \,y \right ) \alpha \right ) }\]
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Added June 26, 2019.
Problem Chapter 7.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta x) w_z = c \sin ^k(\gamma x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*x]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\gamma x)\right )}{\gamma k+\gamma }+c_1\left (z-\frac {b \sqrt {\cos ^2(\beta x)} \sec (\beta x) \sin ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*x)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \sin \left ( x\gamma \right ) \right ) ^{k}\,{\rm d}x+{\it \_F1} \left ( -\int \!a \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \sin \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^m(\beta y) w_z = c \sin ^k(\gamma y)+s \sin ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*x]^m*D[w[x,y,z],z]== c*Sin[gamma*y]^k+s*Sin[mu*z]^r; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \sin ^k\left (\frac {\gamma \left (-a \sqrt {\cos ^2(\lambda x)} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{n+1}(\lambda x)+a \sqrt {\cos ^2(\lambda K[1])} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda K[1])\right ) \sec (\lambda K[1]) \sin ^{n+1}(\lambda K[1])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )+s \sin ^r\left (\frac {\mu \left (-b \sqrt {\cos ^2(\beta x)} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\beta x)\right ) \sec (\beta x) \sin ^{m+1}(\beta x)+b \sqrt {\cos ^2(\beta K[1])} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\beta K[1])\right ) \sec (\beta K[1]) \sin ^{m+1}(\beta K[1])+\beta (m+1) z\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (z-\frac {b \sqrt {\cos ^2(\beta x)} \sec (\beta x) \sin ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*x)^m*diff(w(x,y,z),z)= c*sin(gamma*y)^k+s*sin(mu*z)^r; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \sin \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) +\int ^{x}\!c \left ( \sin \left ( \gamma \, \left ( a\int \! \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{k}+s \left ( \sin \left ( \mu \, \left ( b\int \! \left ( \sin \left ( \beta \,{\it \_f} \right ) \right ) ^{m}\,{\rm d}{\it \_f}-\int \!b \left ( \sin \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \right ) \right ) ^{r}{d{\it \_f}}\]
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Added June 26, 2019.
Problem Chapter 7.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\beta x) w_y + c \sin (\lambda x) w_z = k \sin (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[beta*x]*D[w[x, y,z], y] + c*Sin[lambda*x]*D[w[x,y,z],z]== k*Sin[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \sin \left (\frac {\gamma (a \lambda z+c \cos (\lambda x)-c \cos (\lambda K[1]))}{a \lambda }\right )}{a}dK[1]+c_1\left (\frac {b \cos (\beta x)}{a \beta }+y,\frac {c \cos (\lambda x)}{a \lambda }+z\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*sin(beta*x)*diff(w(x,y,z),y)+ c*sin(lambda*x)*diff(w(x,y,z),z)= k*sin(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya\beta +b\cos \left ( \beta \,x \right ) }{a\beta }},{\frac {za\lambda +c\cos \left ( x\lambda \right ) }{a\lambda }} \right ) -\int ^{x}\!-{\frac {k}{a}\sin \left ( {\frac {\gamma \, \left ( za\lambda +c\cos \left ( x\lambda \right ) -c\cos \left ( {\it \_a}\,\lambda \right ) \right ) }{a\lambda }} \right ) }{d{\it \_a}}\]
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Added June 26, 2019.
Problem Chapter 7.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \sin ^{m_1}(\beta _1 y) w_y + c_1 \sin ^{k_1}(\gamma _1 z) w_z = a_2 \sin ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y)+ c_2 \sin ^{k_2}(\gamma _2 z) \]
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; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*sin(lambda1*x)^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*x)^n2+ b2*sin(beta2*y)^m2+ c2*sin(gamma2*z)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,-\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) +\int ^{x}\!{\frac { \left ( \sin \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( \sin \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+ \left ( \sin \left ( \beta 2\,\RootOf \left ( \int \! \left ( \sin \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \sin \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}{\it b2}+{\it c2}\, \left ( \sin \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \sin \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \sin \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \sin \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_f}}\]
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