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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{k\lambda } \left ( a\sum _{i=0}^{\left \lceil k/2\right \rceil -1} \left ( \prod _{j=1}^{i}{\frac {-k+2\,j-1}{-k+2\,j}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k-2\,i-1} \right ) \cos \left ( \lambda \,x \right ) -k\lambda \, \left ( a\prod _{j=0}^{\left \lceil k/2\right \rceil -1}{\frac {-k+2\,j+1}{-k+2\,j}}x+xb-y \right ) \right ) } \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \sin \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \] 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \sin \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {y\lambda +x}{\lambda }}}\! \left ( 1+a \left ( \sin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \] 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\sqrt {2\,{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) +2} \left ( \left ( \cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a+y \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) +1/2\,\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}\HeunCPrime \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \lambda \right ) \left ( \left ( 4\,\cos \left ( \lambda \,x \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}a+ \left ( \left ( 4\,\sin \left ( \lambda \,x \right ) a+2\,\lambda \right ) \cos \left ( \lambda \,x \right ) +4\,y\sin \left ( \lambda \,x \right ) \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) +4\,y \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) +2\,\HeunCPrime \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\sin \left ( \lambda \,x \right ) +1/2 \right ) \lambda \,\cos \left ( \lambda \,x \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) +{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) \right ) ^{-1}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b+y} \left ( \left ( b+y \right ) \int \!{{\rm e}^{{\frac {-2\,xb\beta -a\cos \left ( \beta \,x \right ) }{\beta }}}}\,{\rm d}x+{{\rm e}^{{\frac {-2\,xb\beta -a\cos \left ( \beta \,x \right ) }{\beta }}}} \right ) } \right ) \] 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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\int _1^x\frac {\exp \left (\frac {a \sin ^{m+1}(b K[1]) \left (\frac {2 b \cos (b K[1]) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\sin ^2(b K[1])\right ) K[1]}{m+1}-2^{-m-1} \sqrt {\pi } \text {Gamma}(m+1) \, _3\tilde {F}_2\left (1,\frac {m+2}{2},\frac {m+2}{2};\frac {m+3}{2},\frac {m+4}{2};\sin ^2(b K[1])\right ) \sin (b K[1])\right )}{2 b^2}\right )}{K[1]^2}dK[1]-\frac {\exp \left (\frac {a \sin ^{m+1}(b x) \left (\frac {2 b x \cos (b x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\sin ^2(b x)\right )}{m+1}-\sqrt {\pi } 2^{-m-1} \text {Gamma}(m+1) \sin (b x) \, _3\tilde {F}_2\left (1,\frac {m+2}{2},\frac {m+2}{2};\frac {m+3}{2},\frac {m+4}{2};\sin ^2(b x)\right )\right )}{2 b^2}\right )}{x^2 y+x}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{xy+1} \left ( xy\int \!{{\rm e}^{\int \!{\frac {a \left ( \sin \left ( xb \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {a \left ( \sin \left ( xb \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac {a \left ( \sin \left ( xb \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\sqrt {\pi } \left ( y+\cos \left ( \lambda \,x \right ) \right ) }{y\erfi \left ( \cos \left ( \lambda \,x \right ) \right ) \sqrt {\pi }+\cos \left ( \lambda \,x \right ) \erfi \left ( \cos \left ( \lambda \,x \right ) \right ) \sqrt {\pi }-2\,{{\rm e}^{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}}}}} \right ) \]
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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 ) ={\it \_F1} \left ( -{\sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2} \left ( \cos \left ( \lambda \,x \right ) \left ( \left ( {a}^{3} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}+ \left ( -2\,{a}^{3}-{a}^{2}\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}+a \left ( a+\lambda \right ) ^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-2\,\lambda \, \left ( a+\lambda \right ) ^{2}\sin \left ( \lambda \,x \right ) + \left ( a+\lambda \right ) ^{2}\lambda \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}+ \left ( -{a}^{3}+{a}^{2}\lambda +a{\lambda }^{2} \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{4}+ \left ( 2\,{a}^{3}-{a}^{2}\lambda -3\,a{\lambda }^{2}-{\lambda }^{3} \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}- \left ( a-\lambda \right ) \left ( a+\lambda \right ) ^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}+2\,\lambda \, \left ( a+\lambda \right ) ^{2}\sin \left ( \lambda \,x \right ) - \left ( a+\lambda \right ) ^{2}\lambda \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}+\sin \left ( \lambda \,x \right ) \left ( \left ( -a \left ( a+2\,\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}+ \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 ) +\sin \left ( \lambda \,x \right ) y \left ( \sin \left ( \lambda \,x \right ) -1 \right ) \left ( \sin \left ( \lambda \,x \right ) a-a-\lambda \right ) ^{2} \right ) a \left ( \cos \left ( \lambda \,x \right ) -1 \right ) \left ( \cos \left ( \lambda \,x \right ) +1 \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) -\sin \left ( \lambda \,x \right ) y \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 ( \sin \left ( \lambda \,x \right ) a-a-\lambda \right ) ^{2} \right ) \left ( \sqrt {\sin \left ( \lambda \,x \right ) +1} \left ( 2\, \left ( \left ( a+\lambda \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) -\sin \left ( \lambda \,x \right ) a \right ) \cos \left ( \lambda \,x \right ) \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{3/2} \left ( \sin \left ( \lambda \,x \right ) a-a-\lambda \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) -1/2 \right ) \lambda \, \left ( \cos \left ( \lambda \,x \right ) -1 \right ) \left ( \cos \left ( \lambda \,x \right ) +1 \right ) + \left ( \sin \left ( \lambda \,x \right ) -1 \right ) ^{5/2}\sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \left ( \left ( {a}^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}- \left ( a+\lambda \right ) a\sin \left ( \lambda \,x \right ) -\lambda \, \left ( a+\lambda \right ) \right ) a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}+ \left ( -{a}^{3}+{a}^{2}\lambda +a{\lambda }^{2} \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}+ \left ( {a}^{3}-2\,a{\lambda }^{2}-{\lambda }^{3} \right ) \sin \left ( \lambda \,x \right ) +a\lambda \, \left ( a+\lambda \right ) \right ) \left ( {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \right ) ^{2}+ \left ( \left ( -\sin \left ( \lambda \,x \right ) {a}^{2}+ \left ( a+\lambda \right ) ^{2} \right ) \cos \left ( \lambda \,x \right ) +\sin \left ( \lambda \,x \right ) y \left ( \sin \left ( \lambda \,x \right ) a-a-\lambda \right ) ^{2} \right ) a \left ( \cos \left ( \lambda \,x \right ) -1 \right ) \left ( \cos \left ( \lambda \,x \right ) +1 \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) -y \left ( \cos \left ( \lambda \,x \right ) -1 \right ) \left ( \cos \left ( \lambda \,x \right ) +1 \right ) \left ( a+\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) a-a-\lambda \right ) ^{2} \right ) \right ) \int ^{\sin \left ( \lambda \,x \right ) }\!{\frac { \left ( {\it \_a}-1 \right ) a-\lambda }{ \left ( {\it \_a}-1 \right ) ^{3/2}\sqrt {{\it \_a}+1}}{{\rm e}^{{\frac {{\it \_a}\,a}{\lambda }}}}}{d{\it \_a}}+2\,{{\rm e}^{{\frac {\sin \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \sin \left ( \lambda \,x \right ) -1 \right ) \left ( \left ( a+\lambda \right ) {\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) -\sin \left ( \lambda \,x \right ) a \right ) \sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \left ( \sin \left ( \lambda \,x \right ) a-a-\lambda \right ) ^{2}{\it csgn} \left ( \sin \left ( \lambda \,x \right ) \right ) \lambda \, \left ( \cos \left ( \lambda \,x \right ) -1 \right ) \left ( \cos \left ( \lambda \,x \right ) +1 \right ) \right ) ^{-1}} \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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1} \left ( \left ( 1/2\,a\sin \left ( 2\,\lambda \,x \right ) +y \left ( \lambda +a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \right ) \right ) \cos \left ( 2\,\lambda \,x \right ) + \left ( -\lambda -a/2 \right ) \sin \left ( 2\,\lambda \,x \right ) -y \left ( \lambda +a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \right ) \right ) \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) \left ( 2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1} \left ( \left ( 1/2\,a\sin \left ( 2\,\lambda \,x \right ) +y \left ( \lambda +a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \right ) \right ) \cos \left ( 2\,\lambda \,x \right ) + \left ( -\lambda -a/2 \right ) \sin \left ( 2\,\lambda \,x \right ) -y \left ( \lambda +a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \right ) \right ) \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) \int \!-2\,{\frac { \left ( a\cos \left ( 2\,\lambda \,x \right ) -a-2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) \lambda }{ \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{3/2}\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}}{{\rm e}^{1/2\,{\frac {a\cos \left ( 2\,\lambda \,x \right ) }{\lambda }}}}}\,{\rm d}x-4\,{{\rm e}^{1/2\,{\frac {a\cos \left ( 2\,\lambda \,x \right ) }{\lambda }}}}\lambda \,\sin \left ( 2\,\lambda \,x \right ) \sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) } \left ( a\cos \left ( 2\,\lambda \,x \right ) -a-2\,\lambda \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{y{x}^{k}x-1} \left ( \left ( -y{x}^{k}x+1 \right ) \int \!-{x}^{k}{{\rm e}^{\int \!{\frac {{x}^{k} \left ( \sin \left ( x \right ) \right ) ^{m}a{x}^{2}-2\,k-2}{x}}\,{\rm d}x}} \left ( k+1 \right ) \,{\rm d}x-{{\rm e}^{\int \!{\frac {{x}^{k} \left ( \sin \left ( x \right ) \right ) ^{m}a{x}^{2}-2\,k-2}{x}}\,{\rm d}x}}{x}^{k}x \right ) } \right ) \]
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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'));
server hangs
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( ba\int \!{x}^{k-1} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{m}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{k+y} \left ( \left ( k+y \right ) \int \!{\frac {1}{a\sin \left ( \lambda \,x \right ) +b}{{\rm e}^{{\frac {1}{\lambda } \left ( c\int \!{\frac {\sin \left ( \mu \,x \right ) }{a\sin \left ( \lambda \,x \right ) +b}}\,{\rm d}x\lambda \,\sqrt {-{a}^{2}+{b}^{2}}-4\,k\arctan \left ( {\frac {b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\cos \left ( 1/2\,\lambda \,x \right ) \sqrt {-{a}^{2}+{b}^{2}}}} \right ) \right ) {\frac {1}{\sqrt {-{a}^{2}+{b}^{2}}}}}}}}\,{\rm d}x+{{\rm e}^{{\frac {1}{\lambda } \left ( c\int \!{\frac {\sin \left ( \mu \,x \right ) }{a\sin \left ( \lambda \,x \right ) +b}}\,{\rm d}x\lambda \,\sqrt {-{a}^{2}+{b}^{2}}-4\,k\arctan \left ( {\frac {b\sin \left ( 1/2\,\lambda \,x \right ) +a\cos \left ( 1/2\,\lambda \,x \right ) }{\cos \left ( 1/2\,\lambda \,x \right ) \sqrt {-{a}^{2}+{b}^{2}}}} \right ) \right ) {\frac {1}{\sqrt {-{a}^{2}+{b}^{2}}}}}}} \right ) } \right ) \]
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