Added June 26, 2019.
Problem Chapter 7.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \cos ^k(\lambda x)+s \]
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*Cos[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-c x)-\frac {c \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*cos(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \cos \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \cos (\beta z) w_z = k \cos (\lambda x)+ s \cos (\gamma y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Cos[beta*z]*D[w[x,y,z],z]== k*Cos[lambda*x]+s*Cos[gamma*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta z) \left (2 \left (2 \sec (\beta z) \sqrt {\sin ^2(\beta z) \cos ^2(\beta z) \sinh ^2\left (\frac {\beta c x}{a}\right ) \left (\cosh \left (\frac {4 \beta c x}{a}\right )-\sinh \left (\frac {4 \beta c x}{a}\right )\right )}+\sinh ^3\left (\frac {\beta c x}{a}\right )+\sinh \left (\frac {\beta c x}{a}\right )\right )-2 \cosh ^3\left (\frac {\beta c x}{a}\right )+\left (1-3 \cosh \left (\frac {2 \beta c x}{a}\right )\right ) \cosh \left (\frac {\beta c x}{a}\right )+6 \sinh \left (\frac {\beta c x}{a}\right ) \cosh ^2\left (\frac {\beta c x}{a}\right )\right )}{4 \cosh \left (\frac {2 \beta c x}{a}\right )-4 \sinh \left (\frac {2 \beta c x}{a}\right )}\right )}{\beta }\right )+\frac {k \sin (\lambda x)}{a \lambda }+\frac {s \sin (\gamma y)}{b \gamma }\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*cos(beta*z)*diff(w(x,y,z),z)= k*cos(lambda*x)+s*cos(gamma*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{b\gamma \,a\lambda } \left ( {\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {a}{\beta \,c}\ln \left ( \RootOf \left ( \beta \,z-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {x\beta \,c}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {x\beta \,c}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {x\beta \,c}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {x\beta \,c}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) b\gamma \,a\lambda +\sin \left ( x\lambda \right ) kb\gamma +sa\sin \left ( \gamma \,y \right ) \lambda \right ) }\]
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Added June 26, 2019.
Problem Chapter 7.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cos ^n(\beta x) w_y + b \cos ^k(\lambda x) w_z = c \cos ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[beta*x]^n*D[w[x, y,z], y] + b*Cos[lambda*x]^k*D[w[x,y,z],z]== c*Cos[gamma*x]^m+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\beta x)\right )}{\beta n+\beta }+y,\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\lambda x)\right )}{k \lambda +\lambda }+z\right )-\frac {c \sqrt {\sin ^2(\gamma x)} \csc (\gamma x) \cos ^{m+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*cos(beta*x)^n*diff(w(x,y,z),y)+ b*cos(lambda*x)^k*diff(w(x,y,z),z)= c*cos(gamma*x)^m+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \cos \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \cos \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cos \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]
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Added June 26, 2019.
Problem Chapter 7.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cos ^n(\lambda x) w_y + b \cos ^m(\beta y) w_z = c \cos ^k(\gamma y)+s \cos ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[lambda*x]^n*D[w[x, y,z], y] + b*Cos[beta*x]^m*D[w[x,y,z],z]== c*Cos[gamma*y]^k+s*Cos[mu*z]^r; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \cos ^k\left (\frac {\gamma \left (a \csc (\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )+s \cos ^r\left (\frac {\mu \left (b \csc (\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\beta x)\right ) \sqrt {\sin ^2(\beta x)} \cos ^{m+1}(\beta x)+\beta (m+1) z-b \cos ^{m+1}(\beta K[1]) \csc (\beta K[1]) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\beta K[1])\right ) \sqrt {\sin ^2(\beta K[1])}\right )}{\beta (m+1)}\right )\right )dK[1]+c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\beta x)\right )}{\beta m+\beta }+z,\frac {a \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda x)\right )}{\lambda n+\lambda }+y\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*cos(lambda*x)^n*diff(w(x,y,z),y)+ b*cos(beta*x)^m*diff(w(x,y,z),z)= c*cos(gamma*y)^k+s*cos(mu*z)^r; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cos \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cos \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) +\int ^{x}\!c \left ( \cos \left ( \gamma \, \left ( a\int \! \left ( \cos \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \cos \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{k}+s \left ( \cos \left ( \mu \, \left ( b\int \! \left ( \cos \left ( \beta \,{\it \_f} \right ) \right ) ^{m}\,{\rm d}{\it \_f}-\int \!b \left ( \cos \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) \right ) \right ) ^{r}{d{\it \_f}}\]
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Added June 26, 2019.
Problem Chapter 7.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cos (\beta x) w_y + c \cos (\lambda x) w_z = k \cos (\gamma z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cos[beta*x]*D[w[x, y,z], y] + c*Cos[lambda*x]*D[w[x,y,z],z]== k*Cos[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \cos \left (\frac {\gamma (a \lambda z-c \sin (\lambda x)+c \sin (\lambda K[1]))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b \sin (\beta x)}{a \beta },z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*cos(beta*x)*diff(w(x,y,z),y)+ c*cos(lambda*x)*diff(w(x,y,z),z)= k*cos(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya\beta -b\sin \left ( \beta \,x \right ) }{a\beta }},{\frac {za\lambda -c\sin \left ( x\lambda \right ) }{a\lambda }} \right ) +\int ^{x}\!{\frac {k}{a}\cos \left ( {\frac {\gamma \, \left ( za\lambda -c\sin \left ( x\lambda \right ) +c\sin \left ( {\it \_a}\,\lambda \right ) \right ) }{a\lambda }} \right ) }{d{\it \_a}}\]
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Added June 26, 2019.
Problem Chapter 7.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \cos ^{n_1}(\lambda _1 x) w_x + b_1 \cos ^{m_1}(\beta _1 y) w_y + c_1 \cos ^{k_1}(\gamma _1 z) w_z = a_2 \cos ^{n_2}(\lambda _2 x) + b_2 \cos ^{m_2}(\beta _2 y)+ c_2 \cos ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Cos[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cos[beta1*y]^m1*D[w[x, y,z], y] + c1*Cos[gamma1*z]^k1*D[w[x,y,z],z]==a2*Cos[lambda2*z]^n2+ b2*Cos[beta2*y]^m2 + c2*Cos[gamma2*z]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*cos(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*cos(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cos(gamma1*z)^k1*diff(w(x,y,z),z)= a2*cos(lambda2*x)^n2+ b2*cos(beta2*y)^m2+ c2*cos(gamma2*z)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \cos \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,-\int \! \left ( \cos \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac {{\it a1}\, \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) +\int ^{x}\!{\frac { \left ( \cos \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( \cos \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+ \left ( \cos \left ( \beta 2\,\RootOf \left ( \int \! \left ( \cos \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \cos \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}{\it b2}+ \left ( \cos \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \cos \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cos \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac {{\it a1}\, \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}{\it c2} \right ) }{d{\it \_f}}\]
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