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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)+\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = s x +\int c \left (\sin ^{k}\left (\lambda x \right )\right )d x +\mathit {\_F1} \left (-a x +y , -b x +z \right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = \frac {a \alpha b \beta \mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \ln \left (\RootOf \left (\gamma z -\arctan \left (\frac {2 \mathit {\_Z} \,{\mathrm e}^{\frac {c \gamma x}{a}}}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 c \gamma x}{a}}+1}, -\frac {\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 c \gamma x}{a}}-1}{\mathit {\_Z}^{2} {\mathrm e}^{\frac {2 c \gamma x}{a}}+1}\right )\right )\right )}{c \gamma }\right )-a \alpha s \cos \left (\beta y \right )-b \beta k \cos \left (\alpha x \right )}{a \alpha b \beta }\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y,z)\to \frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\gamma x)\right )}{\gamma k+\gamma }+c_1\left (z-\frac {b \sqrt {\cos ^2(\beta x)} \sec (\beta x) \sin ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \int c \left (\sin ^{k}\left (\gamma x \right )\right )d x +\mathit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int b \left (\sin ^{m}\left (\beta x \right )\right )d x \right )\right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \left (\sin ^{k}\left (\left (a \left (\int \left (\sin ^{n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )+y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right )\right ) \gamma \right )\right )+s \left (\sin ^{r}\left (\left (b \left (\int \left (\sin ^{m}\left (\mathit {\_f} \beta \right )\right )d \mathit {\_f} \right )+z -\left (\int b \left (\sin ^{m}\left (\beta x \right )\right )d x \right )\right ) \mu \right )\right )\right )d\mathit {\_f} +\mathit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int b \left (\sin ^{m}\left (\beta x \right )\right )d x \right )\right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = -\left (\int _{}^{x}-\frac {k \sin \left (\frac {\left (a \lambda z -c \cos \left (\mathit {\_a} \lambda \right )+c \cos \left (\lambda x \right )\right ) \gamma }{a \lambda }\right )}{a}d\mathit {\_a} \right )+\mathit {\_F1} \left (\frac {a \beta y +b \cos \left (\beta x \right )}{a \beta }, \frac {a \lambda z +c \cos \left (\lambda x \right )}{a \lambda }\right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\left (\mathit {a2} \left (\sin ^{\mathit {n2}}\left (\mathit {\_f} \lambda 2 \right )\right )+\mathit {b2} \left (\sin ^{\mathit {m2}}\left (\beta 2 \RootOf \left (\int \left (\sin ^{-\mathit {n1}}\left (\mathit {\_f} \lambda 1 \right )\right )d \mathit {\_f} -\left (\int \left (\sin ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\sin ^{-\mathit {m1}}\left (\beta 1 y \right )\right )}{\mathit {b1}}d y -\left (\int ^{\mathit {\_Z}}\frac {\mathit {a1} \left (\sin ^{-\mathit {m1}}\left (\mathit {\_a} \beta 1 \right )\right )}{\mathit {b1}}d \mathit {\_a} \right )\right )\right )\right )+\mathit {c2} \left (\sin ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\sin ^{-\mathit {n1}}\left (\mathit {\_f} \lambda 1 \right )\right )d \mathit {\_f} -\left (\int \left (\sin ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\sin ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {a1} \left (\sin ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )\right ) \left (\sin ^{-\mathit {n1}}\left (\mathit {\_f} \lambda 1 \right )\right )}{\mathit {a1}}d\mathit {\_f} +\mathit {\_F1} \left (-\left (\int \left (\sin ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\sin ^{-\mathit {m1}}\left (\beta 1 y \right )\right )}{\mathit {b1}}d y , -\left (\int \left (\sin ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\sin ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z \right )\]
____________________________________________________________________________________