Added January 14, 2019.
Problem 2.6.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a \cos ^k(\lambda x)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Cos[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {a \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(a*cos(lambda*x)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -bx+y-\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x \right ) \] Contains unresolved integral
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Added January 14, 2019.
Problem 2.6.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a \cos ^k(\lambda y)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Cos[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{a \cos ^k(\lambda K[1])+b}dK[1]-x\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(a*cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \cos \left ( \lambda \,y \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]
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Added January 14, 2019.
Problem 2.6.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +a \cos ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Cos[lambda*y]^k*Cos[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\cos ^{-k}(\lambda K[1]) \cos ^{-n}(\mu K[1])dK[1]-a x\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a*cos(lambda*y)^k*cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {xa-\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k} \left ( \cos \left ( \mu \,y \right ) \right ) ^{-n}\,{\rm d}y}{a}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +a \cos ^k(x+\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Cos[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*cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {\lambda \,y+x}{\lambda }}}\! \left ( 1+a \left ( \cos \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \]
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Added January 14, 2019.
Problem 2.6.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2-a^2 + a \lambda \cos (\lambda x) + a^2 \cos ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 - a^2 + a*lambda*Cos[lambda*x] + a^2*Cos[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*cos(lambda*x) + a^2*cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\sqrt {2\,\cos \left ( \lambda \,x \right ) +2} \left ( \left ( a\sin \left ( \lambda \,x \right ) -y \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) +1/2\,\lambda \,\sin \left ( \lambda \,x \right ) \HeunCPrime \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \right ) \left ( \left ( \left ( 4\,\cos \left ( \lambda \,x \right ) a+4\,a+2\,\lambda \right ) \sin \left ( \lambda \,x \right ) -4\,y \left ( \cos \left ( \lambda \,x \right ) +1 \right ) \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) +2\,\lambda \,\sin \left ( \lambda \,x \right ) \HeunCPrime \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \left ( \cos \left ( \lambda \,x \right ) +1 \right ) \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( \lambda \cos (\lambda x) y^2 + \lambda \cos ^3(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*Cos[lambda*x]*y^2 + lambda*Cos[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*cos(lambda*x)*y^2 + lambda*cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( -1-\sin \left ( \lambda \,x \right ) \left ( y-\sin \left ( \lambda \,x \right ) \right ) {{\sl M}\left (1,\,{\frac {3}{2}},\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )} \right ) \left ( -2+\sin \left ( \lambda \,x \right ) \left ( y-\sin \left ( \lambda \,x \right ) \right ) {{\sl U}\left (1,\,{\frac {3}{2}},\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )} \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ 2 w_x +\left ( (\lambda +a+a \cos (\lambda x)) y^2 + \lambda - a + a \cos (\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = 2*D[w[x, y], x] + ((lambda + a + a*Cos[lambda*x])*y^2 + lambda - a + a*Cos[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*cos(lambda*x))*y^2 +lambda - a + a *cos(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y\cos \left ( \lambda \,x \right ) +y-\sin \left ( \lambda \,x \right ) }{\lambda }{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\sqrt {\cos \left ( \lambda \,x \right ) +1}\sqrt {\cos \left ( \lambda \,x \right ) -1} \left ( \left ( y\cos \left ( \lambda \,x \right ) +y-\sin \left ( \lambda \,x \right ) \right ) {{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\sqrt {\cos \left ( \lambda \,x \right ) +1}\sqrt {\cos \left ( \lambda \,x \right ) -1}\int \!{ \left ( \lambda +a+\cos \left ( \lambda \,x \right ) a \right ) \sin \left ( \lambda \,x \right ) {{\rm e}^{-{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}{\frac {1}{\sqrt {\cos \left ( \lambda \,x \right ) -1}}} \left ( \cos \left ( \lambda \,x \right ) +1 \right ) ^{-{\frac {3}{2}}}}\,{\rm d}x+2\,\sin \left ( \lambda \,x \right ) \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( (\lambda +a \cos ^2(\lambda x)) y^2 + \lambda - a + a \cos ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + ((lambda + a*Cos[lambda*x]^2)*y^2 + lambda - a + a*Cos[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*cos(lambda*x)^2)*y^2 + lambda - a + a*cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{\frac { \left ( -1/8\, \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) \left ( a\cos \left ( 2\,\lambda \,x \right ) +a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) +y \left ( \cos \left ( \lambda \,x \right ) \right ) ^{4} \left ( \lambda + \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a \right ) \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) -1}}{\lambda }{{\rm e}^{1/2\,{\frac {a\cos \left ( 2\,\lambda \,x \right ) }{\lambda }}}} \left ( 4\,{{\rm e}^{1/2\,{\frac {a\cos \left ( 2\,\lambda \,x \right ) }{\lambda }}}} \left ( -1/8\, \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) \left ( a\cos \left ( 2\,\lambda \,x \right ) +a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) +y \left ( \cos \left ( \lambda \,x \right ) \right ) ^{4} \left ( \lambda + \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a \right ) \right ) \sqrt {\cos \left ( 2\,\lambda \,x \right ) -1}\int \!{\frac { \left ( a\cos \left ( 2\,\lambda \,x \right ) +a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{\sqrt {\cos \left ( 2\,\lambda \,x \right ) -1} \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{-1/2\,{\frac {a\cos \left ( 2\,\lambda \,x \right ) }{\lambda }}}}}\,{\rm d}x+\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\sin \left ( 2\,\lambda \,x \right ) \left ( a\cos \left ( 2\,\lambda \,x \right ) +a+2\,\lambda \right ) \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m+b x) w_x +\cos ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + b*x)*D[w[x, y], x] + Cos[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 := (a*x^n*y^m+b*x)*diff(w(x,y),x)+ cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y+{x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n +b x \cos ^m y) w_x +y^k w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x*Cos[y]^m)*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n+b*x*cos(y)^m)*diff(w(x,y),x)+y^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y+{x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n +b x \cos ^m y) w_x + \cos ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x*Cos[y]^m)*D[w[x, y], x] + Cos[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 := (a*x^n+b*x*cos(y)^m)*diff(w(x,y),x)+cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y+{x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n \cos ^m y + b x) w_x + \cos ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*Cos[y]^m + b*x)*D[w[x, y], x] + Cos[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 := (a*x^n*cos(y)^m+b*x)*diff(w(x,y),x)+cos(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y+{x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \right ) \]
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