Added January 14, 2019.
Problem 2.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \sin ^k(\lambda x) + b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Sin[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y},Assumptions -> {Element[k, Integers], k > 0}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(a*sin(lambda*x)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming k::integer,k>0),output='realtime'));
\[ \text {Unable to process Latex} \]
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Added January 14, 2019.
Problem 2.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \sin ^k(\lambda y) + b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Sin[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{a \sin ^k(\lambda K[1])+b}dK[1]-x\right )\right \}\right \}\] contains unresolved integral
Maple ✓
restart; pde := diff(w(x,y),x)+(a*sin(lambda*y)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (x -\left (\int \frac {1}{a \left (\sin ^{k}\left (\lambda y \right )\right )+b}d y \right )\right )\] contains unresolved integral
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Added January 14, 2019.
Problem 2.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a \sin ^k(\lambda y) \sin ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Sin[lambda*x]^k*Sin[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\cos ^2(\mu y)} \sec (\mu y) \sin ^{1-n}(\mu y) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(\mu y)\right )}{\mu -\mu n}-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a*sin(lambda*x)^k*sin(mu*y)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\left (\int \left (\sin ^{k}\left (\lambda x \right )\right )d x \right )+\int \frac {\sin ^{-n}\left (\mu y \right )}{a}d y \right )\] contains unresolved integral
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Added January 14, 2019.
Problem 2.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a \sin ^k(x+\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Sin[x + lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+a*sin(x+lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\lambda \left (\int _{}^{\frac {\lambda y +x}{\lambda }}\frac {1}{a \lambda \left (\sin ^{k}\left (\mathit {\_a} \lambda \right )\right )+1}d\mathit {\_a} \right )+x \right )\] contains unresolved integral
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Added January 14, 2019.
Problem 2.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (y^2-a^2 + a \lambda \sin (\lambda x)+a^2 \sin ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 - a^2 + a*lambda*Sin[lambda*x] + a^2*Sin[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+(y^2-a^2 + a*lambda*sin(lambda*x)+a^2*sin(lambda*x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {2 \,\mathrm {csgn}\left (\sin \left (\lambda x \right )\right ) \sin \left (\lambda x \right )+2}\, \left (\frac {\lambda \HeunCPrime \left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )^{2}}{2}+\left (a \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )^{2}+y \right ) \HeunC \left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{2 \left (\mathrm {csgn}\left (\sin \left (\lambda x \right )\right )+\sin \left (\lambda x \right )\right ) \lambda \HeunCPrime \left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )+\left (4 a \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )^{2}+4 y +\left (4 y \sin \left (\lambda x \right )+\left (4 a \sin \left (\lambda x \right )+2 \lambda \right ) \cos \left (\lambda x \right )\right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )\right ) \HeunC \left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}\right )\]
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Added January 14, 2019.
Problem 2.6.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2 + a \sin (\beta x) y + a b \sin (\beta x)-b^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*Sin[beta*x]*y + a*b*Sin[beta*x] - b^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+( y^2 + a*sin(beta*x)* y + a*b*sin(beta*x)-b^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (b +y \right ) \left (\int {\mathrm e}^{\frac {-2 b \beta x -a \cos \left (\beta x \right )}{\beta }}d x \right )+{\mathrm e}^{\frac {-2 b \beta x -a \cos \left (\beta x \right )}{\beta }}}{b +y}\right )\] contains unresolved integrals
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Added January 14, 2019.
Problem 2.6.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2 + a x \sin ^m(b x) y + a \sin ^m(b x)\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*x*Sin[b*x]^m*y + a*Sin[b*x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\int _1^x\frac {\exp \left (\frac {a \sin ^{m+1}(b K[1]) \left (\frac {2 b \cos (b K[1]) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\sin ^2(b K[1])\right ) K[1]}{m+1}-2^{-m-1} \sqrt {\pi } \operatorname {Gamma}(m+1) \, _3\tilde {F}_2\left (1,\frac {m+2}{2},\frac {m+2}{2};\frac {m+3}{2},\frac {m+4}{2};\sin ^2(b K[1])\right ) \sin (b K[1])\right )}{2 b^2}\right )}{K[1]^2}dK[1]-\frac {\exp \left (\frac {a \sin ^{m+1}(b x) \left (\frac {2 b x \cos (b x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\sin ^2(b x)\right )}{m+1}-\sqrt {\pi } 2^{-m-1} \sin (b x) \operatorname {Gamma}(m+1) \, _3\tilde {F}_2\left (1,\frac {m+2}{2},\frac {m+2}{2};\frac {m+3}{2},\frac {m+4}{2};\sin ^2(b x)\right )\right )}{2 b^2}\right )}{x^2 y+x}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+( y^2 + a*x*sin(b*x)^m*y + a*sin(b*x)^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {x y \left (\int {\mathrm e}^{\int \frac {a x^{2} \left (\sin ^{m}\left (b x \right )\right )-2}{x}d x}d x \right )+x \,{\mathrm e}^{\int \frac {a x^{2} \left (\sin ^{m}\left (b x \right )\right )-2}{x}d x}+\int {\mathrm e}^{\int \frac {a x^{2} \left (\sin ^{m}\left (b x \right )\right )-2}{x}d x}d x}{x y +1}\right )\]
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Added January 14, 2019.
Problem 2.6.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left (\lambda \sin (\lambda x) y^2 + \lambda \sin ^3(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + lambda*Sin[lambda*x]^3)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+(lambda*sin(lambda*x)*y^2 + lambda*sin(lambda*x)^3)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\sqrt {\pi }\, \left (y +\cos \left (\lambda x \right )\right )}{\sqrt {\pi }\, y \erfi \left (\cos \left (\lambda x \right )\right )+\sqrt {\pi }\, \erfi \left (\cos \left (\lambda x \right )\right ) \cos \left (\lambda x \right )-2 \,{\mathrm e}^{\cos ^{2}\left (\lambda x \right )}}\right )\]
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Added January 14, 2019.
Problem 2.6.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ 2 w_x +\left ((\lambda +a-a \sin (\lambda x)) y^2 + \lambda -a -a \sin (\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = 2*D[w[x, y], x] + ((lambda + a - a*Sin[lambda*x])*y^2 + lambda - a - a*Sin[lambda*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := 2*diff(w(x,y),x)+((lambda+a-a*sin(lambda*x))*y^2 +lambda -a -a*sin(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (\left (\cos \left (\lambda x \right )-1\right ) \left (\cos \left (\lambda x \right )+1\right ) \left (\left (\sin \left (\lambda x \right )-1\right ) \left (a \sin \left (\lambda x \right )-a -\lambda \right )^{2} y \sin \left (\lambda x \right )+\left (-\left (a +2 \lambda \right ) a \left (\sin ^{2}\left (\lambda x \right )\right )+\left (2 a^{2}+5 a \lambda +3 \lambda ^{2}\right ) \sin \left (\lambda x \right )-\left (a +2 \lambda \right ) \left (a +\lambda \right )\right ) \cos \left (\lambda x \right )\right ) a \,\mathrm {csgn}\left (\sin \left (\lambda x \right )\right ) \sin \left (\lambda x \right )+\left (\left (-a^{3}+\lambda a^{2}+a \lambda ^{2}\right ) \left (\sin ^{4}\left (\lambda x \right )\right )+\left (2 a^{3}-\lambda a^{2}-3 a \lambda ^{2}-\lambda ^{3}\right ) \left (\sin ^{3}\left (\lambda x \right )\right )+2 \left (a +\lambda \right )^{2} \lambda \sin \left (\lambda x \right )+\left (a^{3} \left (\sin ^{4}\left (\lambda x \right )\right )+\left (a +\lambda \right )^{2} a \left (\sin ^{2}\left (\lambda x \right )\right )+\left (-2 a^{3}-\lambda a^{2}\right ) \left (\sin ^{3}\left (\lambda x \right )\right )-2 \left (a +\lambda \right )^{2} \lambda \sin \left (\lambda x \right )+\left (a +\lambda \right )^{2} \lambda \right ) \left (\cos ^{2}\left (\lambda x \right )\right )-\left (a -\lambda \right ) \left (a +\lambda \right )^{2} \left (\sin ^{2}\left (\lambda x \right )\right )-\left (a +\lambda \right )^{2} \lambda \right ) \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )^{2}-\left (\cos \left (\lambda x \right )-1\right ) \left (\cos \left (\lambda x \right )+1\right ) \left (\sin \left (\lambda x \right )-1\right ) \left (a +\lambda \right ) \left (a \sin \left (\lambda x \right )-a -\lambda \right )^{2} y \sin \left (\lambda x \right )\right ) \left (\sin \left (\lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\sin \left (\lambda x \right )+1}}{2 \left (-a \sin \left (\lambda x \right )+\left (a +\lambda \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )\right ) \left (a \sin \left (\lambda x \right )-a -\lambda \right )^{2} \left (\cos \left (\lambda x \right )-1\right ) \left (\cos \left (\lambda x \right )+1\right ) \left (\sin \left (\lambda x \right )-1\right ) \lambda \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right ) {\mathrm e}^{\frac {a \sin \left (\lambda x \right )}{\lambda }} \sin \left (\lambda x \right )+\left (2 \left (\sin \left (\lambda x \right )-1\right )^{\frac {3}{2}} \left (-a \sin \left (\lambda x \right )+\left (a +\lambda \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )\right ) \left (a \sin \left (\lambda x \right )-a -\lambda \right ) \left (\cos \left (\lambda x \right )-1\right ) \left (\cos \left (\lambda x \right )+1\right ) \left (\sin \left (\lambda x \right )-\frac {1}{2}\right ) \lambda \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )+\left (\left (\left (a^{2} \left (\sin ^{2}\left (\lambda x \right )\right )-\left (a +\lambda \right ) a \sin \left (\lambda x \right )-\left (a +\lambda \right ) \lambda \right ) a \left (\cos ^{2}\left (\lambda x \right )\right )+\left (a +\lambda \right ) a \lambda +\left (-a^{3}+\lambda a^{2}+a \lambda ^{2}\right ) \left (\sin ^{2}\left (\lambda x \right )\right )+\left (a^{3}-2 a \lambda ^{2}-\lambda ^{3}\right ) \sin \left (\lambda x \right )\right ) \cos \left (\lambda x \right ) \mathrm {csgn}\left (\sin \left (\lambda x \right )\right )^{2}+\left (\left (a \sin \left (\lambda x \right )-a -\lambda \right )^{2} y \sin \left (\lambda x \right )+\left (-a^{2} \sin \left (\lambda x \right )+\left (a +\lambda \right )^{2}\right ) \cos \left (\lambda x \right )\right ) \left (\cos \left (\lambda x \right )-1\right ) \left (\cos \left (\lambda x \right )+1\right ) a \,\mathrm {csgn}\left (\sin \left (\lambda x \right )\right )-\left (\cos \left (\lambda x \right )-1\right ) \left (\cos \left (\lambda x \right )+1\right ) \left (a +\lambda \right ) \left (a \sin \left (\lambda x \right )-a -\lambda \right )^{2} y \right ) \left (\sin \left (\lambda x \right )-1\right )^{\frac {5}{2}} \sin \left (\lambda x \right )\right ) \sqrt {\sin \left (\lambda x \right )+1}\, \left (\int _{}^{\sin \left (\lambda x \right )}\frac {\left (\left (\mathit {\_a} -1\right ) a -\lambda \right ) {\mathrm e}^{\frac {\mathit {\_a} a}{\lambda }}}{\left (\mathit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\mathit {\_a} +1}}d\mathit {\_a} \right )}\right )\]
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Added January 14, 2019.
Problem 2.6.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ((\lambda +a \sin ^2(\lambda x)) y^2 + \lambda -a +a \sin ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + ((lambda + a*Sin[lambda*x]^2)*y^2 + lambda - a + a*Sin[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+((lambda+a*sin(lambda*x)^2)*y^2 + lambda -a +a*sin(lambda*x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {\cos \left (2 \lambda x \right )+1}\, \left (-\left (a \left (\sin ^{2}\left (\lambda x \right )\right )+\lambda \right ) y +\left (\frac {a \sin \left (2 \lambda x \right )}{2}+\left (a \left (\sin ^{2}\left (\lambda x \right )\right )+\lambda \right ) y \right ) \cos \left (2 \lambda x \right )+\left (-\frac {a}{2}-\lambda \right ) \sin \left (2 \lambda x \right )\right ) \left (\cos \left (2 \lambda x \right )-1\right )}{-4 \sqrt {\cos \left (2 \lambda x \right )-1}\, \left (a \cos \left (2 \lambda x \right )-a -2 \lambda \right ) \lambda \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right )+2 \sqrt {\cos \left (2 \lambda x \right )+1}\, \left (-\left (a \left (\sin ^{2}\left (\lambda x \right )\right )+\lambda \right ) y +\left (\frac {a \sin \left (2 \lambda x \right )}{2}+\left (a \left (\sin ^{2}\left (\lambda x \right )\right )+\lambda \right ) y \right ) \cos \left (2 \lambda x \right )+\left (-\frac {a}{2}-\lambda \right ) \sin \left (2 \lambda x \right )\right ) \left (\cos \left (2 \lambda x \right )-1\right ) \left (\int -\frac {2 \left (a \cos \left (2 \lambda x \right )-a -2 \lambda \right ) \lambda \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right )}{\left (\cos \left (2 \lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\cos \left (2 \lambda x \right )+1}}d x \right )}\right )\]
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Added January 14, 2019.
Problem 2.6.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x -\left ( (k+1) x^k y^2 - a x^{k+1}(\sin x)^m y + a (\sin x)^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - a*x^(k + 1)*Sin[x]^m*y + a*Sin[x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)-((k+1)*x^k*y^2 - a*x^(k+1)*(sin(x))^m*y + a*(sin(x))^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-x y x^{k} \left (\int \left (k +1\right ) x^{k} {\mathrm e}^{\int \frac {a x^{2} x^{k} \left (\sin ^{m}x \right )-2 k -2}{x}d x}d x \right )+x x^{k} {\mathrm e}^{\int \frac {a x^{2} x^{k} \left (\sin ^{m}x \right )-2 k -2}{x}d x}+\int \left (k +1\right ) x^{k} {\mathrm e}^{\int \frac {a x^{2} x^{k} \left (\sin ^{m}x \right )-2 k -2}{x}d x}d x}{x y x^{k}-1}\right )\]
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Added January 14, 2019.
Problem 2.6.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a \sin ^k(\lambda x + \mu )(y-b x^n -c)^2 + y - b x^n + b n x^{n-1} - c \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Sin[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+(a*sin(lambda*x + mu)^k * (y-b*x^n -c)^2 + y - b*x^n + b*n*x^(n-1) - c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
time expired
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Added January 14, 2019.
Problem 2.6.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x +\left ( a \sin ^m(\lambda x ) y^2 + k y + a b^2 x^{2 k} \sin ^m(\lambda x) \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*Sin[lambda*x]^m*y^2 + k*y + a*b^2*x^(2*k)*Sin[lambda*x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^xa K[1]^{k-1} \sin ^m(\lambda K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+(a*sin(lambda*x)^m*y^2 + k*y + a*b^2*x^(2*k)*sin(lambda*x)^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (a b \left (\int x^{k -1} \left (\sin ^{m}\left (\lambda x \right )\right )d x \right )-\arctan \left (\frac {y x^{-k}}{b}\right )\right )\]
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Added January 14, 2019.
Problem 2.6.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a \sin (\lambda x) + b) w_x +\left ( y^2+ c \sin (\mu x) y - k^2 + c k \sin (\mu x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*Sin[lambda*x] + b)*D[w[x, y], x] + (y^2 + c*Sin[mu*x]*y - k^2 + c*k*Sin[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a *sin(lambda*x) + b)*diff(w(x,y),x)+(y^2+ c*sin(mu*x)* y - k^2 + c*k*sin(mu*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (k +y \right ) \left (\int \frac {{\mathrm e}^{\frac {\sqrt {-a^{2}+b^{2}}\, c \lambda \left (\int \frac {\sin \left (\mu x \right )}{a \sin \left (\lambda x \right )+b}d x \right )-4 k \arctan \left (\frac {a \cos \left (\frac {\lambda x}{2}\right )+b \sin \left (\frac {\lambda x}{2}\right )}{\sqrt {-a^{2}+b^{2}}\, \cos \left (\frac {\lambda x}{2}\right )}\right )}{\sqrt {-a^{2}+b^{2}}\, \lambda }}}{a \sin \left (\lambda x \right )+b}d x \right )+{\mathrm e}^{\frac {\sqrt {-a^{2}+b^{2}}\, c \lambda \left (\int \frac {\sin \left (\mu x \right )}{a \sin \left (\lambda x \right )+b}d x \right )-4 k \arctan \left (\frac {a \cos \left (\frac {\lambda x}{2}\right )+b \sin \left (\frac {\lambda x}{2}\right )}{\sqrt {-a^{2}+b^{2}}\, \cos \left (\frac {\lambda x}{2}\right )}\right )}{\sqrt {-a^{2}+b^{2}}\, \lambda }}}{k +y}\right )\]
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