Added January 20, 2019.
Problem 2.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \sin ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Sin[lambda*x]^k*Cos[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\sin ^2(\mu y)} \csc (\mu y) \cos ^{1-n}(\mu y) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(\mu y)\right )}{\mu (n-1)}-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ a*sin(lambda*x)^k*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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] Has unresolved integrals
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Added January 20, 2019.
Problem 2.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-y \tan x+a(1-a) \cot ^2 x \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 - y*Tan[x] + a*(1 - a)*Cot[x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4}} \left (i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4} \cos (x)+2 y \sin (x)+\cos (x)\right )}{-i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4} \cos (x)+2 y \sin (x)+\cos (x)}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (y^2-y *tan(x)+a*(1-a)*cot(x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( \sin \left ( x \right ) \right ) ^{-1+2\,a} \left ( y\sin \left ( x \right ) +a\cos \left ( x \right ) \right ) }{ \left ( a-1 \right ) \cos \left ( x \right ) -y\sin \left ( x \right ) }} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-m y \tan x+b^2 \cos ^{2 m} x \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 - m*y*Tan[x] + b^2*Cos[x]^(2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {b^2} \sqrt {\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(x)\right )}{m+1}+\tan ^{-1}\left (\frac {y \cos ^{-m}(x)}{\sqrt {b^2}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (y^2-m*y*tan(x)+b^2*cos(x)^(2*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 ) ={\it \_F1} \left ( { \left ( 3\,b \left ( \cos \left ( x \right ) \right ) ^{2} \left ( 1/3\, \left ( \cos \left ( x \right ) -1 \right ) \left ( \cos \left ( x \right ) +1 \right ) \left ( m-1 \right ) \hypergeom \left ( [3/2,-m/2+3/2],[5/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) +\hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) +3\,y \left ( \cos \left ( x \right ) \right ) ^{m}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \right ) \left ( -3\,y \left ( \cos \left ( x \right ) \right ) ^{m}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) +3\,b \left ( \cos \left ( x \right ) \right ) ^{2}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \left ( 1/3\, \left ( \cos \left ( x \right ) -1 \right ) \left ( \cos \left ( x \right ) +1 \right ) \left ( m-1 \right ) \hypergeom \left ( [3/2,-m/2+3/2],[5/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) +\hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \right ) ^{-1}} \right ) \] Mathematica answer is simpler
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Added January 20, 2019.
Problem 2.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+m y \cot x+b^2 \sin ^m x \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + m*y*Cot[x] + b^2*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)+ (y^2+m*y*cot(x)+b^2*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'));
server hangs Server hangs
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Added January 20, 2019.
Problem 2.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-2 \lambda ^2 \tan ^2(\lambda x)-2 \lambda ^2 \cot ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 - 2*lambda^2*Tan[lambda*x]^2 - 2*lambda^2*Cot[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^2-2*lambda^2*tan(lambda*x)^2-2*lambda^2*cot(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -4\,{\frac {\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \left ( y\sin \left ( 2\,\lambda \,x \right ) -2\,\cos \left ( 2\,\lambda \,x \right ) \lambda \right ) }{-4\,y\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) }\sin \left ( 2\,\lambda \,x \right ) \ln \left ( \cos \left ( \lambda \,x \right ) +1/2\,\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \right ) +8\,\cos \left ( 2\,\lambda \,x \right ) \sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) }\ln \left ( \cos \left ( \lambda \,x \right ) +\sqrt { \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}-1} \right ) \lambda +y \left ( \sin \left ( 5\,\lambda \,x \right ) +\sin \left ( 3\,\lambda \,x \right ) \right ) \cos \left ( 2\,\lambda \,x \right ) -y\sin \left ( 3\,\lambda \,x \right ) -y\sin \left ( 5\,\lambda \,x \right ) +7\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) -\cos \left ( 3\,\lambda \,x \right ) -1/7\,\cos \left ( 5\,\lambda \,x \right ) +1/7\,\cos \left ( 7\,\lambda \,x \right ) \right ) }} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+\lambda (a+b)+2 a b+a(\lambda -a) \tan ^2(\lambda x)+ b(\lambda -b) \cot ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda*(a + b) + 2*a*b + a*(lambda - a)*Tan[lambda*x]^2 + b*(lambda - b)*Cot[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+lambda*(a+b)+2*a*b+a*(lambda -a)*tan(lambda*x)^2+ b*(lambda -b)*cot(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{ \left ( \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a- \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}b-\sin \left ( \lambda \,x \right ) y\cos \left ( \lambda \,x \right ) \right ) \left ( -3/2\,\lambda +a \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {a}{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {b}{\lambda }}} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {a-\lambda }{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {b-\lambda }{\lambda }}} \left ( -4\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( a+b-\lambda \right ) \hypergeom \left ( [2,{\frac {-a-b+2\,\lambda }{\lambda }}],[-1/2\,{\frac {-5\,\lambda +2\,a}{\lambda }}], \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2} \right ) +2\,\hypergeom \left ( [1,{\frac {-b+\lambda -a}{\lambda }}],[-1/2\,{\frac {-3\,\lambda +2\,a}{\lambda }}], \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2} \right ) \left ( \left ( \lambda -b \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}+\sin \left ( \lambda \,x \right ) y\cos \left ( \lambda \,x \right ) + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( a-\lambda \right ) \right ) \left ( -3/2\,\lambda +a \right ) \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \cos ^n(\lambda x) y-a \cos ^{n-1}(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Cos[lambda*x]^n*y - a*Cos[lambda*x]^(n - 1))*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 + a*cos(lambda*x)^n*y-a*cos(lambda*x)^(n-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
server hangs
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Added January 20, 2019.
Problem 2.6.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \sin (\lambda x) y-a \tan (\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Sin[lambda*x]*y - a*Tan[lambda*x])*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 + a*sin(lambda*x)*y-a*tan(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{(\cos \left ( \lambda \,x \right ) y-1){{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( a{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y-1 \right ) \Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) -y\lambda \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \sin (\lambda x) y-a \tan (\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Sin[lambda*x]*y - a*Tan[lambda*x])*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 + a*sin(lambda*x)*y-a*tan(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{(y\cos \left ( \lambda \,x \right ) -1){{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( a{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( y\cos \left ( \lambda \,x \right ) -1 \right ) \Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) -y\lambda \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( A e^{\lambda x} \cos (a y) + B e^{\mu x} \sin (a y) + A e^{\lambda x} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (A*Exp[lambda*x]*Cos[a*y] + B*Exp[mu*x]*Sin[a*y] + A*Exp[lambda*x])*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*exp(lambda*x)*cos(a*y) + B*exp(mu*x)*sin(a*y) + A*exp(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{2\,a \left ( -\lambda +\mu \right ) } \left ( Aa \left ( \cos \left ( ay \right ) +1 \right ) \int \!{{\rm e}^{{\frac {-a{{\rm e}^{\mu \,x}}B+\lambda \,x\mu }{\mu }}}}\,{\rm d}x-{{\rm e}^{-{\frac {a{{\rm e}^{\mu \,x}}B}{\mu }}}}\sin \left ( ay \right ) \right ) \left ( \cos \left ( {\frac {ay}{2}} \right ) \right ) ^{-2}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \sin ^{n+1}(2 x) w_x + \left ( a y^2 \sin ^{2 n}x + b \cos ^{2 n} x \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = Sin[2*x]^(n + 1)*D[w[x, y], x] + (a*y^2*Sin[x]^(2*n) + b*Cos[x]^(2*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✓
restart; pde := sin(2*x)^(n+1)*diff(w(x,y),x)+ (a*y^2*sin(x)^(2*n) + b*cos(x)^(2*n) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{ \left ( \left ( \sin \left ( x \right ) \right ) ^{\sqrt {{n}^{2}-a{4}^{-n}b}+2\,n}ya+ \left ( \sin \left ( 2\,x \right ) \right ) ^{n} \left ( \sin \left ( x \right ) \right ) ^{\sqrt {{n}^{2}-a{4}^{-n}b}} \left ( n+\sqrt {{n}^{2}-a{4}^{-n}b} \right ) \right ) \left ( \cos \left ( x \right ) \right ) ^{-\sqrt {{n}^{2}-a{4}^{-n}b}} \left ( ay \left ( \sin \left ( x \right ) \right ) ^{2\,n}- \left ( \sin \left ( 2\,x \right ) \right ) ^{n}\sqrt {{n}^{2}-a{4}^{-n}b}+ \left ( \sin \left ( 2\,x \right ) \right ) ^{n}n \right ) ^{-1}} \right ) \]
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