Added January 2, 2019.
Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a y + b x^k \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y + b*x^k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (b a^{-k-1} \text {Gamma}(k+1,a x)+y e^{-a x}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*y+b*x^k)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -{{\rm e}^{1/2\,ax}}{x}^{k} \left ( ax \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,ax \right ) b+ya \left ( k+1 \right ) \right ) {{\rm e}^{-ax}}}{a \left ( k+1 \right ) }} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^k y+b x^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^k*y + b*x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[\left \{\left \{w(x,y)\to c_1\left (b (k+1)^{\frac {n-k}{k+1}} a^{-\frac {n+1}{k+1}} \text {Gamma}\left (\frac {n+1}{k+1},\frac {a x^{k+1}}{k+1}\right )+y e^{-\frac {a x^{k+1}}{k+1}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*x^k*y+b*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a \left ( 2\,k+n+3 \right ) \left ( n+1 \right ) \left ( k+n+2 \right ) } \left ( - \left ( {\frac {{x}^{k+1}a}{k+1}} \right ) ^{{\frac {-k-n-2}{2\,k+2}}}b \left ( \left ( k+n+2 \right ) {x}^{-k+n}+a{x}^{n+1} \right ) {{\rm e}^{-{\frac {{x}^{k+1}a}{2\,k+2}}}} \left ( k+1 \right ) ^{2} \WhittakerM \left ( {\frac {-k+n}{2\,k+2}},{\frac {2\,k+n+3}{2\,k+2}},{\frac {{x}^{k+1}a}{k+1}} \right ) +2\, \left ( -1/2\,{{\rm e}^{-{\frac {{x}^{k+1}a}{2\,k+2}}}}{x}^{-k+n} \left ( {\frac {{x}^{k+1}a}{k+1}} \right ) ^{{\frac {-k-n-2}{2\,k+2}}}b \left ( k+1 \right ) \left ( k+n+2 \right ) \WhittakerM \left ( {\frac {k+n+2}{2\,k+2}},{\frac {2\,k+n+3}{2\,k+2}},{\frac {{x}^{k+1}a}{k+1}} \right ) +{{\rm e}^{-{\frac {{x}^{k+1}a}{k+1}}}} \left ( n+1 \right ) \left ( k+n/2+3/2 \right ) ay \right ) \left ( k+n+2 \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a y^2+b x^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y^2 + b*x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol=Simplify[sol];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {-2 a x y \text {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \text {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}{(2 a x y+1) \text {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \left (\text {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\text {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*y^2+b*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( -\BesselY \left ( {\frac {n+3}{n+2}},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \sqrt {ab}{x}^{n/2}x+y\BesselY \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) ax+\BesselY \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \right ) \left ( \sqrt {ab}{x}^{n/2}x\BesselJ \left ( {\frac {n+3}{n+2}},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) -\BesselJ \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) yax-\BesselJ \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \right ) ^{-1} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+a n x^{n-1} -a^2 x^{2 n} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*n*x^(n - 1) - a^2*x^(2*n))*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*n*x^(n-1)-a^2*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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{({x}^{5/2\,n+2}a-{x}^{3/2\,n+2}y){{\rm e}^{-2\,{\frac {a{x}^{n+1}}{n+1}}}} \left ( -3\, \left ( a \left ( n+4/3 \right ) {x}^{n+1}+1/3\, \left ( n+2 \right ) \left ( -xy+n+1 \right ) \right ) \left ( n+2 \right ) {{\rm e}^{-{\frac {a{x}^{n+1}}{n+1}}}} \WhittakerM \left ( {\frac {n+2}{2\,n+2}},{\frac {2\,n+3}{2\,n+2}},-2\,{\frac {a{x}^{n+1}}{n+1}} \right ) +2\, \left ( {a}^{2}{x}^{2\,n+2}-1/2\,a \left ( n+2 \right ) {x}^{n+1}-ay{x}^{n+2}-1/2\, \left ( n+2 \right ) \left ( -xy+n+1 \right ) \right ) \left ( n+1 \right ) {{\rm e}^{-{\frac {a{x}^{n+1}}{n+1}}}} \WhittakerM \left ( -{\frac {n}{2\,n+2}},{\frac {2\,n+3}{2\,n+2}},-2\,{\frac {a{x}^{n+1}}{n+1}} \right ) +2\, \left ( -2\,{\frac {a{x}^{n+1}}{n+1}} \right ) ^{{\frac {3\,n+4}{2\,n+2}}} \left ( n+2 \right ) ^{2} \left ( n+3/2 \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2 + a x^n y + a x^{n-1} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*x^n*y + a*x^(n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[\left \{\left \{w(x,y)\to c_1\left ((-1)^{\frac {1}{n+1}} (n+1)^{-\frac {n+2}{n+1}} a^{\frac {1}{n+1}} \text {Gamma}\left (-\frac {1}{n+1},-\frac {a x^{n+1}}{n+1}\right )-\frac {e^{\frac {a x^{n+1}}{n+1}}}{x^2 y+x}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (y^2+a*x^n*y+a*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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{anx \left ( xy+1 \right ) \left ( 1+2\,n \right ) } \left ( - \left ( -{\frac {a{x}^{n+1}}{n+1}} \right ) ^{-{\frac {n}{2\,n+2}}}{{\rm e}^{{\frac {a{x}^{n+1}}{2\,n+2}}}} \left ( n+1 \right ) ^{2} \left ( axy-{x}^{-n}ny-n{x}^{-n-1}+a \right ) \WhittakerM \left ( {\frac {-n-2}{2\,n+2}},{\frac {1+2\,n}{2\,n+2}},-{\frac {a{x}^{n+1}}{n+1}} \right ) +2\, \left ( 1/2\, \left ( -{\frac {a{x}^{n+1}}{n+1}} \right ) ^{-{\frac {n}{2\,n+2}}}n{{\rm e}^{{\frac {a{x}^{n+1}}{2\,n+2}}}} \left ( {x}^{-n}y+{x}^{-n-1} \right ) \left ( n+1 \right ) \WhittakerM \left ( {\frac {n}{2\,n+2}},{\frac {1+2\,n}{2\,n+2}},-{\frac {a{x}^{n+1}}{n+1}} \right ) +a{{\rm e}^{{\frac {a{x}^{n+1}}{n+1}}}} \left ( n+1/2 \right ) \right ) n \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+a x^n y-a b x^n -b^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*x^n*y - a*b*x^n - 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*x^n*y-a*b*x^n-b^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b-y} \left ( \left ( -b+y \right ) \int \!{{\rm e}^{{\frac {x \left ( {x}^{n}a+2\,b \left ( n+1 \right ) \right ) }{n+1}}}}\,{\rm d}x+{{\rm e}^{{\frac {x \left ( {x}^{n}a+2\,b \left ( n+1 \right ) \right ) }{n+1}}}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2 + b x^{-n-2} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*x^(-n - 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{\sqrt {(n+1)^2-4 a b}} \left (\sqrt {(n+1)^2-4 a b}+2 a y x^{n+1}+n+1\right )}{\sqrt {(n+1)^2-4 a b}-2 a y x^{n+1}-n-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*x^n*y^2+b*x^(-n-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ab-{n}^{2}-2\,n-1}} \left ( \ln \left ( x \right ) \sqrt {4\,ab-{n}^{2}-2\,n-1}-2\,\arctan \left ( {\frac {2\,a{x}^{n}yx+n+1}{\sqrt {4\,ab-{n}^{2}-2\,n-1}}} \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2 + b m x^{m-1} -a b^2 x^{n+2 m} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*m*x^(m - 1) - a*b^2*x^(n + 2*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)+ (a*x^n*y^2 + b*m*x^(m-1) -a*b^2*x^(n+2*m))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{a \left ( -{x}^{5/2\,m+2\,n+2}b+{x}^{3/2\,m+2\,n+2}y \right ) {{\rm e}^{-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}}}} \left ( -3\, \left ( m+2\,n+2 \right ) {{\rm e}^{-{\frac {{x}^{m+n+1}ab}{m+n+1}}}} \left ( ab \left ( m+4/3\,n+4/3 \right ) {x}^{m+n+1}-1/3\, \left ( m+2\,n+2 \right ) \left ( ay{x}^{n+1}-m-n-1 \right ) \right ) \WhittakerM \left ( {\frac {m+2\,n+2}{2\,n+2\,m+2}},{\frac {2\,m+3\,n+3}{2\,n+2\,m+2}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +2\, \left ( {a}^{2}{x}^{2\,n+2\,m+2}{b}^{2}-{a}^{2}by{x}^{m+2\,n+2}-1/2\, \left ( m+2\,n+2 \right ) \left ( {x}^{m+n+1}ab-ay{x}^{n+1}+m+n+1 \right ) \right ) \left ( m+n+1 \right ) {{\rm e}^{-{\frac {{x}^{m+n+1}ab}{m+n+1}}}} \WhittakerM \left ( -{\frac {m}{2\,n+2\,m+2}},{\frac {2\,m+3\,n+3}{2\,n+2\,m+2}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +2\, \left ( m+3/2\,n+3/2 \right ) \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{{\frac {3\,m+4\,n+4}{2\,n+2\,m+2}}} \left ( m+2\,n+2 \right ) ^{2} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( (n+1)x^n y^2 - a x^{n+m+1} y + a x^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + ((n + 1)*x^n*y^2 - a*x^(n + m + 1)*y + a*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)+ ((n+1)*x^n*y^2 - a*x^(n+m+1)* y + a*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {m+2\,n+3}{m+1} \left ( ax{x}^{m} \left ( -n+m \right ) \hypergeom \left ( [{\frac {2+2\,m}{m+n+2}}],[{\frac {2\,m+3+n}{m+n+2}}],{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}} \right ) -\hypergeom \left ( [{\frac {-n+m}{m+n+2}}],[{\frac {m+1}{m+n+2}}],{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}} \right ) \left ( {x}^{m}xa-y \left ( n+1 \right ) \right ) \left ( m+1 \right ) \right ) \left ( a{x}^{2}{x}^{m}{x}^{n} \left ( m+1 \right ) \hypergeom \left ( [{\frac {2\,m+3+n}{m+n+2}}],[{\frac {2\,m+3\,n+5}{m+n+2}}],{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}} \right ) - \left ( m+2\,n+3 \right ) \hypergeom \left ( [{\frac {m+1}{m+n+2}}],[{\frac {m+2\,n+3}{m+n+2}}],{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}} \right ) \left ( x \left ( {x}^{m}xa-y \left ( n+1 \right ) \right ) {x}^{n}-n-1 \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2 + b x^m y+ b c x^m -a c^2 x^n \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*x^m*y + b*c*x^m - a*c^2*x^n)*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*x^n*y^2 + b*x^m*y+ b*c*x^m -a*c^2*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{c+y} \left ( \left ( -c-y \right ) \int \!{x}^{n}a{{\rm e}^{{\frac {b \left ( n+1 \right ) {x}^{m+1}-2\,a{x}^{n+1}c \left ( m+1 \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}}\,{\rm d}x-{{\rm e}^{{\frac {b \left ( n+1 \right ) {x}^{m+1}-2\,a{x}^{n+1}c \left ( m+1 \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2-a x^n (b x^m +c) y+ b m x^{m-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 - a*x^n*(b*x^m + c)*y + b*m*x^(m - 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)+ (a*x^n*y^2-a*x^n*(b*x^m +c)*y+ b*m*x^(m-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x - \left (a n x^{n-1} y^2 - c x^m (a x^n+b) + c x^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] - (a*n*x^(n - 1)*y^2 - c*x^m*(a*x^n + b) + c*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)- (a*n*x^(n-1)*y^2 - c*x^m*(a*x^n+b) + c*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2+b x^m y+ c k x^{k-1}-b c x^{m+k}-a c^2 x^{n+2 k} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*x^m*y + c*k*x^(k - 1) - b*c*x^(m + k) - a*c^2*x^(n + 2*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*x^n*y^2+b*x^m*y+ c*k*x^(k-1)-b*c*x^(m+k)-a*c^2*x^(n+2*k))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^{2 n+1} y^3 + b x^{-n-2} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^(2*n + 1)*y^3 + b*x^(-n - 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)+ (a*x^(2*n+1)*y^3 + b*x^(-n-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \ln \left ( x \right ) -\sum _{{\it \_R}=\RootOf \left ( a{{\it \_Z}}^{3}+{\it \_Z}\, \left ( n+1 \right ) +b \right ) }{\frac {\ln \left ( {x}^{n}yx-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}a+n+1}} \right ) \] Solution contains RootOf
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Added January 2, 2019.
Problem 2.2.5.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2 - b m x^{m-1} - 2 a b^3 x^{n+3 m} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^3 + 3*a*b*x^(n + m)*y^2 - b*m*x^(m - 1) - 2*a*b^3*x^(n + 3*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {6^{-\frac {n+1}{2 m+n+1}} (2 m+n+1)^{-\frac {2 m}{2 m+n+1}} b^{-\frac {2 (n+1)}{2 m+n+1}} e^{-\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \left (6^{\frac {n+1}{2 m+n+1}} (2 m+n+1)^{\frac {2 m}{2 m+n+1}} b^{\frac {2 (n+1)}{2 m+n+1}}-2 a^{\frac {2 m}{2 m+n+1}} \left (b x^m+y\right )^2 e^{\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text {Gamma}\left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )\right )}{\left (b x^m+y\right )^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*x^n*y^3 + 3*a*b*x^(n+m)*y^2 - b*m*x^(m-1) - 2*a*b^3*x^(n+3*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}{{b}^{2} \left ( 4\,m+3\,n+3 \right ) \left ( n+1 \right ) \left ( m+n+1 \right ) \left ( {x}^{2\,m}{b}^{2}+2\,b{x}^{m}y+{y}^{2} \right ) } \left ( \left ( \left ( 4\, \left ( {m}^{3}+ \left ( 5/2\,n+5/2 \right ) {m}^{2}+2\, \left ( n+1 \right ) ^{2}m+3/2\,{n}^{2}+3/2\,n \right ) {y}^{2}{x}^{-2\,m}+4\,b \left ( 2\, \left ( m+n+1 \right ) ^{2} \left ( m+n/2+1/2 \right ) y{x}^{-m}+b \left ( {m}^{3}+ \left ( 5/2\,n+5/2 \right ) {m}^{2}+2\, \left ( n+1 \right ) ^{2}m+3/2\,{n}^{2}+3/2\,n \right ) \right ) \right ) {2}^{{\frac {-m-n-1}{2\,m+n+1}}}+{2}^{{\frac {m}{2\,m+n+1}}} \left ( n+1 \right ) \left ( {n}^{2}-n+1 \right ) \left ( {y}^{2}{x}^{-2\,m}+{b}^{2} \right ) \right ) \left ( {\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}} \right ) ^{{\frac {-m-n-1}{2\,m+n+1}}}{3}^{{\frac {-3\,m-2\,n-2}{2\,m+n+1}}}{{\rm e}^{-3\,{\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},{\frac {4\,m+3\,n+3}{4\,m+2\,n+2}},6\,{\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}} \right ) + \left ( {\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}} \right ) ^{{\frac {-m-n-1}{2\,m+n+1}}} \left ( \left ( \left ( 12\,{b}^{4} \left ( {m}^{2}+ \left ( n+1 \right ) m+n/2 \right ) a{x}^{2\,m+n+1}+24\,{b}^{3} \left ( m+n/2+1/2 \right ) ^{2}ay{x}^{m+n+1}+4\, \left ( {m}^{3}+ \left ( 2\,n+2 \right ) {m}^{2}+5/4\, \left ( n+1 \right ) ^{2}m+1/4\,{n}^{3}+1/4 \right ) {y}^{2}{x}^{-2\,m}+12\,b \left ( 2/3\,y \left ( {m}^{3}+ \left ( 2\,n+2 \right ) {m}^{2}+5/4\, \left ( n+1 \right ) ^{2}m+3/4\,{n}^{2}+3/4\,n \right ) {x}^{-m}+b \left ( \left ( {m}^{2}+ \left ( n+1 \right ) m+n/2 \right ) a{y}^{2}{x}^{n+1}+1/3\,{m}^{3}+ \left ( 2/3\,n+2/3 \right ) {m}^{2}+{\frac {5\, \left ( n+1 \right ) ^{2}m}{12}}+1/12\,{n}^{3}+1/12 \right ) \right ) \right ) {2}^{{\frac {-m-n-1}{2\,m+n+1}}}+{2}^{{\frac {m}{2\,m+n+1}}}by{x}^{-m} \left ( n+1 \right ) \left ( {n}^{2}-n+1 \right ) \right ) {3}^{{\frac {-3\,m-2\,n-2}{2\,m+n+1}}}+{3}^{{\frac {-m-n-1}{2\,m+n+1}}}{2}^{{\frac {-m-n-1}{2\,m+n+1}}} \left ( a{b}^{4} \left ( {n}^{2}+1 \right ) {x}^{2\,m+n+1}+n{y}^{2} \left ( n+1 \right ) {x}^{-2\,m}+{b}^{2} \left ( a{y}^{2} \left ( {n}^{2}+1 \right ) {x}^{n+1}+{n}^{2}+n \right ) \right ) \right ) {{\rm e}^{-3\,{\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},{\frac {4\,m+3\,n+3}{4\,m+2\,n+2}},6\,{\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}} \right ) +4\,{b}^{2} \left ( m+n+1 \right ) {{\rm e}^{-6\,{\frac {{x}^{2\,m+n+1}a{b}^{2}}{2\,m+n+1}}}} \left ( m+3/4\,n+3/4 \right ) \left ( n+1 \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2+ c x^k y- 2 a b^3 x^{n+3 m} + b c x^{m+l} - b m x^{m-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^3 + 3*a*b*x^(n + m)*y^2 + c*x^k*y - 2*a*b^3*x^(n + 3*m) + b*c*x^(m + l) - b*m*x^(m - 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)+ (a*x^n*y^3 + 3*a*b*x^(n+m)*y^2+ c*x^k*y-2*a*b^3*x^(n+3*m) + b*c*x^(m+l)-b*m*x^(m-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 2, 2019.
Problem 2.2.5.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a y^n + b x ^{\frac {n}{1-n}} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y^n + b*x^(n/(1 - n)))*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*y^n+b*x^(n/(1-n)))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int _{{\it \_b}}^{y}\!{x}^{{\frac {n}{n-1}}} \left ( \left ( ax \left ( n-1 \right ) {{\it \_a}}^{n}+{\it \_a} \right ) {x}^{{\frac {n}{n-1}}}+bx \left ( n-1 \right ) \right ) ^{-1}\,{\rm d}{\it \_a}n+\ln \left ( x \right ) +\int _{{\it \_b}}^{y}\!{x}^{{\frac {n}{n-1}}} \left ( \left ( ax \left ( n-1 \right ) {{\it \_a}}^{n}+{\it \_a} \right ) {x}^{{\frac {n}{n-1}}}+bx \left ( n-1 \right ) \right ) ^{-1}\,{\rm d}{\it \_a} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^{m-n-(m n)} y^n + b x^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^(m - n - m*n)*y^n + b*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)+ (a*x^(m-n-(m*n))*y^n + 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \int _{{\it \_b}}^{y}\!-{\frac {{x}^{mn}{x}^{n}}{{x}^{n} \left ( {x}^{m}bx- \left ( m+1 \right ) {\it \_a} \right ) {x}^{mn}+{{\it \_a}}^{n}{x}^{m}xa}}\,{\rm d}{\it \_a}+\ln \left ( x \right ) \right ) \]
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Added January 2, 2019.
Problem 2.2.5.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^k + b x^m y \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*x^n*y^k + b*x^m*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (a (-1)^{\frac {m-n}{m+1}} (m+1)^{\frac {n-m}{m+1}} b^{-\frac {n+1}{m+1}} (k-1)^{\frac {m-n}{m+1}} \text {Gamma}\left (\frac {n+1}{m+1},-\frac {b (k-1) x^{m+1}}{m+1}\right )+y^{1-k} e^{\frac {b (k-1) x^{m+1}}{m+1}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*x^n*y^k + b*x^m*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b \left ( n+1 \right ) \left ( m+n+2 \right ) \left ( 2\,m+3+n \right ) } \left ( -{y}^{{\frac {k}{m+1}}} \left ( -{\frac { \left ( k-1 \right ) {x}^{m+1}b}{m+1}} \right ) ^{{\frac {-m-n-2}{2+2\,m}}}{{\rm e}^{{\frac {b{x}^{m+1}}{m+1}}}} \left ( m+1 \right ) ^{2}{y}^{{\frac {km}{m+1}}} \left ( \left ( m+n+2 \right ) {x}^{n-m}-b{x}^{n+1} \left ( k-1 \right ) \right ) a{{\rm e}^{{\frac { \left ( k-1 \right ) {x}^{m+1}b}{2+2\,m}}}} \WhittakerM \left ( {\frac {n-m}{2+2\,m}},{\frac {2\,m+3+n}{2+2\,m}},-{\frac { \left ( k-1 \right ) {x}^{m+1}b}{m+1}} \right ) - \left ( a{x}^{n-m}{y}^{{\frac {k}{m+1}}}{y}^{{\frac {km}{m+1}}}{{\rm e}^{{\frac {b{x}^{m+1}}{m+1}}}}{{\rm e}^{{\frac { \left ( k-1 \right ) {x}^{m+1}b}{2+2\,m}}}} \left ( -{\frac { \left ( k-1 \right ) {x}^{m+1}b}{m+1}} \right ) ^{{\frac {-m-n-2}{2+2\,m}}} \left ( m+1 \right ) \left ( m+n+2 \right ) \WhittakerM \left ( {\frac {m+n+2}{2+2\,m}},{\frac {2\,m+3+n}{2+2\,m}},-{\frac { \left ( k-1 \right ) {x}^{m+1}b}{m+1}} \right ) -2\,{y}^{{\frac {m}{m+1}}}b{{\rm e}^{{\frac {b{x}^{m+1}k}{m+1}}}}{y}^{ \left ( m+1 \right ) ^{-1}} \left ( m+n/2+3/2 \right ) \left ( n+1 \right ) \right ) \left ( m+n+2 \right ) \right ) \left ( {y}^{{\frac {km}{m+1}}} \right ) ^{-1} \left ( {y}^{{\frac {k}{m+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{x}^{m+1}}{m+1}}}} \right ) ^{-1}} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a y^2 + b y+ c x^{2 b} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*y^2 + b*y + c*x^(2*b))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {a} y \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )+\sqrt {c} x^b \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}{\sqrt {c} x^b \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )-\sqrt {a} y \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*y^2 + b*y+ c*x^(2*b))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b} \left ( {x}^{b}\sqrt {a}\sqrt {c}-\arctan \left ( {\frac {y{x}^{-b}\sqrt {a}}{\sqrt {c}}} \right ) b \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a y^2+(n+b x^n) y + c x^{2 n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*y^2 + (n + b*x^n)*y + c*x^(2*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {x^n \sqrt {b^2-4 a c}}{n}} \left (x^n \sqrt {b^2-4 a c}+2 a y+b x^n\right )}{x^n \sqrt {b^2-4 a c}-2 a y-b x^n}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*y^2+(n+b*x^n)*y + c*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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}}n} \left ( -2\,bn\arctan \left ( {\frac {2\,bay{x}^{-n}+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +{x}^{n}\sqrt {4\,ac{b}^{2}-{b}^{4}} \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a x^n y^2 + b y+ c x^{-n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^n*y^2 + b*y + c/x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {x^{\sqrt {-4 a c+b^2+2 b n+n^2}} \left (\sqrt {-4 a c+b^2+2 b n+n^2}+2 a y x^n+b+n\right )}{-\sqrt {-4 a c+b^2+2 b n+n^2}+2 a y x^n+b+n}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*x^n*y^2+b*y+c*x^(-n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ca-{b}^{2}-2\,bn-{n}^{2}}} \left ( \ln \left ( x \right ) \sqrt {4\,ca-{b}^{2}-2\,bn-{n}^{2}}-2\,\arctan \left ( {\frac {2\,a{x}^{n}y+b+n}{\sqrt {4\,ca-{b}^{2}-2\,bn-{n}^{2}}}} \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a x^n y^2+ m y- a b^2 x^{x+2 m} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^n*y^2 + m*y - a*b^2*x^(x + 2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*x^n*y^2+ m*y- a*b^2*x^(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( \left ( -2\,a{x}^{n}y-m-n \right ) \BesselI \left ( {\frac {-n-m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) -2\,x{\frac {\partial }{\partial x}}\BesselI \left ( {\frac {-n-m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) \right ) \left ( \left ( 2\,a{x}^{n}y+m+n \right ) \BesselK \left ( {\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) +2\,x{\frac {\partial }{\partial x}}\BesselK \left ( {\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) \right ) ^{-1} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( x^{2 n} y^2+(m-n) y+ x^{2 m} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (x^(2*n)*y^2 + (m - n)*y + x^(2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (y x^{n-m}\right )-\frac {x^{m+n}}{m+n}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (x^(2*n)*y^2+(m-n)*y+ 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -n-m \right ) \arctan \left ( {x}^{n-m}y \right ) +{x}^{n+m}}{n+m}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a x^{2 n} y^2+ (b x^n -n) y + c \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^(2*n)*y^2 + (b*x^n - n)*y + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {x^n \sqrt {b^2-4 a c}}{n}} \left (\sqrt {b^2-4 a c}+2 a y x^n+b\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*x^(2*n)*y^2+ (b*x^n -n)*y + c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}}n} \left ( -2\,bn\arctan \left ( {\frac {2\,bay{x}^{n}+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +{x}^{n}\sqrt {4\,ac{b}^{2}-{b}^{4}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a x^{2 n + m} y^2 +(b x^{n+m}-n) y+ c x^m \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^(2*n + m)*y^2 + (b*x^(n + m) - n)*y + c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {\sqrt {b^2-4 a c} x^{m+n}}{m+n}} \left (\sqrt {b^2-4 a c}+2 a y x^n+b\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*x^(2*n + m)*y^2 +(b*x^(n+m)-n)*y+ c*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}} \left ( n+m \right ) } \left ( -2\,b \left ( n+m \right ) \arctan \left ( {\frac {2\,bay{x}^{n}+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +{x}^{n}\sqrt {4\,ac{b}^{2}-{b}^{4}}{x}^{m} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a y^3+3 a b x^n y^2 - b n x^n -2 a b^3 x^{3 n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*y^3 + 3*a*b*x^n*y^2 - b*n*x^n - 2*a*b^3*x^(3*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {3 a b^2 x^{2 n}}{n}} \left (a e^{\frac {3 a b^2 x^{2 n}}{n}} \left (b x^n+y\right )^2 \text {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )+n\right )}{n \left (b x^n+y\right )^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*y^3+3*a*b*x^n*y^2 - b*n*x^n -2*a*b^3*x^(3*n) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{n \left ( {b}^{2}{x}^{2\,n}+2\,{x}^{n}by+{y}^{2} \right ) } \left ( -a \left ( {b}^{2}{x}^{2\,n}+2\,{x}^{n}by+{y}^{2} \right ) \Ei \left ( 1,3\,{\frac {a{x}^{2\,n}{b}^{2}}{n}} \right ) +{{\rm e}^{-3\,{\frac {a{x}^{2\,n}{b}^{2}}{n}}}}n \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a x^{2 n +1} y^3 + (b x -n) y + c x^{1-n} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^(2*n + 1)*y^3 + (b*x - n)*y + c*x^(1 - n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*x^(2*n +1)*y^3 + (b*x-n)*y + c*x^(1-n) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {b}^{3}\sum _{{\it \_R}=\RootOf \left ( {c}^{2}a{{\it \_Z}}^{3}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( {\frac {-{x}^{n}by-{\it \_R}\,c}{c}} \right ) }-bx \right ) \] Solution contains RootOf
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Added January 2, 2019.
Problem 2.2.5.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( a x^{n+2} y^3+ (b x^n-1) y + c x^{n-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*x^(n + 2)*y^3 + (b*x^n - 1)*y + c*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 := x*diff(w(x,y),x)+ (a*x^(n+2)*y^3+ (b*x^n-1)*y + c*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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{n} \left ( \sum _{{\it \_R}=\RootOf \left ( {c}^{2}a{{\it \_Z}}^{3}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( {\frac {-bxy-{\it \_R}\,c}{c}} \right ) }{b}^{2}n-{x}^{n} \right ) } \right ) \] Solution contains RootOf
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Added January 2, 2019.
Problem 2.2.5.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( y+a x^{n - m }y^m+b x^{n-k} y^k \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (y + a*x^(n - m)*y^m + b*x^(n - k)*y^k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := x*diff(w(x,y),x)+ ( y+a*x^(n - m)*y^m+b*x^(n-k)*y^k )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( n-1 \right ) x} \left ( \left ( n-1 \right ) x\int _{{\it \_b}}^{y}\!-{\frac {{x}^{m}{x}^{k}}{x \left ( a{{\it \_a}}^{m}{x}^{k}+{{\it \_a}}^{k}{x}^{m}b \right ) }}\,{\rm d}{\it \_a}+{x}^{n} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ y w_x + \left (x^{n-1}((1+2 n)x+a n) y-n x^{2 n}(x+a) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = y*D[w[x, y], x] + (x^(n - 1)*((1 + 2*n)*x + a*n)*y - n*x^(2*n)*(x + a))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := y*diff(w(x,y),x)+ ( x^(n-1)*((1+2*n)*x+a*n)*y-n*x^(2*n)*(x+a) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {1}{x} \left ( -\sqrt {-{n}^{2}}\int ^{-2\,{\frac {1}{\sqrt {-{n}^{2}}}\arctan \left ( {\frac {n \left ( 2\,{x}^{n}a+{x}^{n+1}-y \right ) }{\sqrt {-{n}^{2}} \left ( {x}^{n+1}-y \right ) }} \right ) }}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt {-{n}^{2}} \right ) {{\rm e}^{-{\it \_a}}}{d{\it \_a}}x+2\,n \left ( a+x/2 \right ) {{\rm e}^{2\,{\frac {1}{\sqrt {-{n}^{2}}}\arctan \left ( {\frac {n \left ( 2\,{x}^{n}a+{x}^{n+1}-y \right ) }{\sqrt {-{n}^{2}} \left ( {x}^{n+1}-y \right ) }} \right ) }}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ y w_x + \left ( (a(2 n +k)x^k+b)x^{n-1}y -(a^2 n x^{2 k}+ a b x^k -c) x^{2 n-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = y*D[w[x, y], x] + ((a*(2*n + k)*x^k + b)*x^(n - 1)*y - (a^2*n*x^(2*k) + a*b*x^k - c)*x^(2*n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✗
restart; pde := y*diff(w(x,y),x)+ ( (a*(2*n+k)*x^k+b)*x^(n-1)*y -(a^2*n*x^(2*k)+ a*b*x^k-c)*x^(2*n-1) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 2, 2019.
Problem 2.2.5.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(2 a x y+b) w_x - \left ( a(m+3) x y^2+b(m+2)y-c x^m \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*(2*a*x*y + b)*D[w[x, y], x] - (a*(m + 3)*x*y^2 + b*(m + 2)*y - c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{m+2} \left (2 (m+1) y (a x y+b)-c x^m\right )}{2 a (m+1)}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*(2*a*x*y+b)*diff(w(x,y),x)- ( a*(m+3)*x*y^2+b*(m+2)*y-c*x^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\frac {{x}^{m}{x}^{2} \left ( -1/2\,c{x}^{m}+y \left ( m+1 \right ) \left ( axy+b \right ) \right ) }{2+2\,m}} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x^2(2 a x y+b) w_x - \left ( 4 a x^2 y^2 + 3 b x y-c x^2 - k \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x^2*(2*a*x*y + b)*D[w[x, y], x] - (4*a*x^2*y^2 + 3*b*x*y - c*x^2 - k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^2 (x (4 y (a x y+b)-c x)-2 k)}{4 a}\right )\right \}\right \}\]
Maple ✓
restart; pde := x^2*(2*a*x*y+b)*diff(w(x,y),x)- ( 4*a*x^2*y^2 + 3*b*x*y-c*x^2 - k )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -a{x}^{4}{y}^{2}-b{x}^{3}y+1/4\,c{x}^{4}+1/2\,k{x}^{2} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^m w_x + b y^n w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^m*D[w[x, y], x] + b*y^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-m}}{a (m-1)}-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^m*diff(w(x,y),x)+ b*y^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{-m+1}b \left ( n-1 \right ) +a{y}^{-n+1} \left ( m-1 \right ) }{a \left ( m-1 \right ) }} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x + (b y+ c x^m) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + (b*y + c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y e^{\frac {b x^{1-n}}{a (n-1)}}-\frac {c (a-a n)^{\frac {-m+n-1}{n-1}} b^{\frac {m-n+1}{n-1}} \text {Gamma}\left (\frac {-m+n-1}{n-1},\frac {b x^{1-n}}{a-a n}\right )}{a (n-1)}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x)+ (b*y+c*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ab \left ( m-3\,n+3 \right ) \left ( m-2\,n+2 \right ) \left ( -n+m+1 \right ) } \left ( a{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{{\frac {m-2\,n+2}{2\,n-2}}}c{x}^{m} \left ( n-1 \right ) \left ( m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {2\,n-2-m}{2\,n-2}},{\frac {-m+3\,n-3}{2\,n-2}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) -{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{{\frac {m-2\,n+2}{2\,n-2}}} \left ( n-1 \right ) ^{2}c \left ( b{x}^{-n+m+1}+a{x}^{m} \left ( m-2\,n+2 \right ) \right ) \WhittakerM \left ( -{\frac {m}{2\,n-2}},{\frac {-m+3\,n-3}{2\,n-2}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) +y{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}ab \left ( m-3\,n+3 \right ) \left ( m-2\,n+2 \right ) \left ( -n+m+1 \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.37 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^k w_x + (y^n+ b x^m y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^k*D[w[x, y], x] + (y^n + b*x^m*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> {n != 1}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left ((k-m-1)^{\frac {m}{k-m-1}} a^{\frac {m}{k-m-1}} b^{\frac {1-k}{k-m-1}} (n-1)^{\frac {m}{-k+m+1}} \text {Gamma}\left (\frac {k-1}{k-m-1},\frac {b (n-1) x^{-k+m+1}}{a (k-m-1)}\right )+y^{1-n} e^{-\frac {b (n-1) x^{-k+m+1}}{a (k-m-1)}}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^k*diff(w(x,y),x)+ (y^n+b*x^m*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n<>1),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a \left ( 3\,k-3-2\,m \right ) b{y}^{ \left ( k-m-1 \right ) ^{-1}} \left ( 2\,k-2-m \right ) \left ( k-1 \right ) } \left ( -4\,{y}^{{\frac {kn}{k-m-1}}} \left ( {\frac {{x}^{-k+m+1}b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {-2\,k+m+2}{2\,k-2\,m-2}}}{y}^{ \left ( k-m-1 \right ) ^{-1}}{{\rm e}^{{\frac {b{x}^{-k+m+1}n}{ \left ( k-m-1 \right ) a}}}} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {k-1}{k-m-1}}} \left ( k-m-1 \right ) \left ( k-1-m/2 \right ) ^{2}{x}^{-m}a{y}^{{\frac {m}{k-m-1}}} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {1-k}{k-m-1}}}{{\rm e}^{-1/2\,{\frac {{x}^{-k+m+1}b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ( {\frac {2\,k-2-m}{2\,k-2\,m-2}},{\frac {3\,k-3-2\,m}{2\,k-2\,m-2}},{\frac {{x}^{-k+m+1}b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) -2\,{y}^{{\frac {kn}{k-m-1}}} \left ( {\frac {{x}^{-k+m+1}b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {-2\,k+m+2}{2\,k-2\,m-2}}}{y}^{ \left ( k-m-1 \right ) ^{-1}}{{\rm e}^{{\frac {b{x}^{-k+m+1}n}{ \left ( k-m-1 \right ) a}}}} \left ( 1/2\,{x}^{-k+m+1}b \left ( n-1 \right ) +a \left ( k-1-m/2 \right ) \right ) \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {k-1}{k-m-1}}} \left ( k-m-1 \right ) ^{2}{x}^{-m}{y}^{{\frac {m}{k-m-1}}} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {1-k}{k-m-1}}}{{\rm e}^{-1/2\,{\frac {{x}^{-k+m+1}b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ( {\frac {m}{2\,k-2\,m-2}},{\frac {3\,k-3-2\,m}{2\,k-2\,m-2}},{\frac {{x}^{-k+m+1}b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) +6\, \left ( k-1-2/3\,m \right ) {y}^{{\frac {n}{k-m-1}}}b{y}^{{\frac {k}{k-m-1}}} \left ( k-1-m/2 \right ) {{\rm e}^{{\frac {b{x}^{-k+m+1}}{ \left ( k-m-1 \right ) a}}}} \left ( k-1 \right ) {y}^{{\frac {mn}{k-m-1}}}a \right ) \left ( {y}^{{\frac {kn}{k-m-1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{x}^{-k+m+1}n}{ \left ( k-m-1 \right ) a}}}} \right ) ^{-1} \left ( {y}^{{\frac {m}{k-m-1}}} \right ) ^{-1}} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.38 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(a x^k+b) w_x + \left ( \alpha x^n y^2+(\beta -a n x^k)y+\gamma x^{-n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*(a*x^k + b)*D[w[x, y], x] + (alpha*x^n*y^2 + (beta - a*n*x^k)*y + gamma/x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+2 \alpha y x^n+b n+\beta \right ) \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}}{b k}\right )}{-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+2 \alpha y x^n+b n+\beta }\right )\right \}\right \}\]
Maple ✓
restart; pde :=x*(a*x^k+b)*diff(w(x,y),x)+ (alpha*x^n*y^2+(beta-a*n*x^k)*y+g*x^(-n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {bn+\beta }{\sqrt { \left ( bn+\beta \right ) ^{2} \left ( {b}^{2}{n}^{2}+2\,b\beta \,n-4\,\alpha \,g+{\beta }^{2} \right ) }bk} \left ( 2\,bk \left ( bn+\beta \right ) \arctanh \left ( {\frac { \left ( bn+\beta \right ) \left ( 2\,\alpha \,{x}^{n}y+bn+\beta \right ) }{\sqrt { \left ( bn+\beta \right ) ^{2} \left ( {b}^{2}{n}^{2}+2\,b\beta \,n-4\,\alpha \,g+{\beta }^{2} \right ) }}} \right ) +\sqrt { \left ( bn+\beta \right ) ^{2} \left ( {b}^{2}{n}^{2}+2\,b\beta \,n-4\,\alpha \,g+{\beta }^{2} \right ) } \left ( k\ln \left ( x \right ) -\ln \left ( {x}^{k}a+b \right ) \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.39 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (y+ A x^n + a) w_x - \left ( n A x^{n-1} y + k x^m + b\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = (y + A*x^n + a)*D[w[x, y], x] - (n*A*x^(n - 1)*y + k*x^m + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y \left (2 a+2 A x^n+y\right )+2 b x+\frac {2 k x^{m+1}}{m+1}\right )\right \}\right \}\]
Maple ✓
restart; pde :=(y+ A*x^n + a)*diff(w(x,y),x)- ( n*A*x^(n-1)*y + k*x^m + b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-2\,k{x}^{m}x-2\, \left ( A{x}^{n}y+ya+bx+1/2\,{y}^{2} \right ) \left ( m+1 \right ) }{2+2\,m}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.40 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (y+ a x^{n+1}+b x^n) w_x + \left (a n x^n + c x^{n-1} \right ) y w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (y + a*x^(n + 1) + b*x^n)*D[w[x, y], x] + (a*n*x^n + c*x^(n - 1))*y*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✗
restart; pde :=(y+ a*x^(n+1)+b*x^n)*diff(w(x,y),x)+ ( a*n*x^n + c*x^(n-1))*y*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.41 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(2 a x^n y+b) w_x - \left (a(3 n+m)x^n y^2+b(2 n+m)y-A x^m -C x^{-n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*(2*a*x^n*y + b)*D[w[x, y], x] - (a*(3*n + m)*x^n*y^2 + b*(2*n + m)*y - A*x^m - C0/x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{m+n} \left (x^n \left (2 y (m+n) \left (a y x^n+b\right )-A x^m\right )-2 \text {C0}\right )}{2 a (m+n)}\right )\right \}\right \}\]
Maple ✓
restart; pde :=x*(2*a*x^n*y+b)*diff(w(x,y),x)- ( a*(3*n+m)*x^n*y^2+b*(2*n+m)*y-A*x^m -C*x^(-n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{x}^{2\,n+2\,m}A-2\,by \left ( n+m \right ) {x}^{m+2\,n}-2\,a{y}^{2} \left ( n+m \right ) {x}^{3\,n+m}+2\,{x}^{n+m}C}{2\,n+2\,m}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.42 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n+b x^2+ x y) w_x + \left (c x^n + b x y+ y^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x^2 + x*y)*D[w[x, y], x] + (c*x^n + b*x*y + y^2)*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^2+ x*y)*diff(w(x,y),x)+ ( c*x^n + b*x*y+ y^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac {{n}^{2}-3\,n+3}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( \ln \left ( 9\,{\frac { \left ( {a}^{2} \left ( n-1 \right ) {x}^{n}+ \left ( \left ( \left ( bx+y \right ) n-bx-2\,y \right ) a+cx \right ) x \right ) \left ( {n}^{2}-3\,n+3 \right ) }{ \left ( 2\,n-3 \right ) \left ( {x}^{n}a+b{x}^{2}+xy \right ) a}} \right ) + \left ( -n+1 \right ) \ln \left ( 9\,{\frac { \left ( {a}^{2}{x}^{n}+{x}^{2} \left ( ab+c \right ) \right ) \left ( {n}^{2}-3\,n+3 \right ) }{ \left ( {x}^{n}a+x \left ( bx+y \right ) \right ) a \left ( n-3 \right ) }} \right ) + \left ( n-2 \right ) \ln \left ( -9\,{\frac { \left ( {n}^{2}-3\,n+3 \right ) \left ( ya-cx \right ) x}{n \left ( {x}^{n}a+x \left ( bx+y \right ) \right ) a}} \right ) -2\, \left ( n-1 \right ) \left ( \ln \left ( x \right ) -1/2\,\ln \left ( {a}^{2}{x}^{n}+{x}^{2} \left ( ab+c \right ) \right ) \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.43 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a y^n+b x^2+c x y) w_x + \left (k y^n+ b x y+c y^2\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*y^n + b*x^2 + c*x*y)*D[w[x, y], x] + (k*y^n + b*x*y + c*y^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde :=(a*y^n+b*x^2+c*x*y)*diff(w(x,y),x)+ ( k*y^n+ b*x*y+c*y^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac {{n}^{2}-3\,n+3}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( \ln \left ( 9/2\,{\frac { \left ( {n}^{2}-3\,n+3 \right ) \left ( {k}^{2} \left ( n-1 \right ) {y}^{n}+y \left ( \left ( c \left ( n-1 \right ) y+bx \left ( n-2 \right ) \right ) k+aby \right ) \right ) }{k \left ( k{y}^{n}+bxy+c{y}^{2} \right ) \left ( n-3/2 \right ) }} \right ) + \left ( -n+1 \right ) \ln \left ( 9\,{\frac { \left ( {y}^{n}{k}^{2}+{y}^{2} \left ( ab+ck \right ) \right ) \left ( {n}^{2}-3\,n+3 \right ) }{ \left ( n-3 \right ) \left ( k{y}^{n}+bxy+c{y}^{2} \right ) k}} \right ) + \left ( n-2 \right ) \ln \left ( 9\,{\frac { \left ( {n}^{2}-3\,n+3 \right ) \left ( ya-kx \right ) by}{k \left ( k{y}^{n}+bxy+c{y}^{2} \right ) n}} \right ) + \left ( n-1 \right ) \left ( \ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ) -2\,\ln \left ( y \right ) \right ) \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.44 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left (c y^2-b x^{m-1} y+ a x^{n-2}\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + (c*y^2 - b*x^(m - 1)*y + a*x^(n - 2))*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^m + c)*diff(w(x,y),x)+ ( c*y^2-b*x^(m-1)*y+ a*x^(n-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 2, 2019.
Problem 2.2.5.45 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left (a x^{n-2} y^2 + b x^{m-1} y + c \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + (a*x^(n - 2)*y^2 + b*x^(m - 1)*y + c)*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^m + c)*diff(w(x,y),x)+ ( a*x^(n-2)*y^2 + b*x^(m-1)*y + c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.46 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left ( \alpha x^k y^2 + \beta x^s y - \alpha \lambda ^2 x^k + \beta \lambda x^s \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + (alpha*x^k*y^2 + beta*x^s*y - alpha*lambda^2*x^k + beta*lambda*x^s)*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^m + c)*diff(w(x,y),x)+ (alpha*x^k*y^2 + beta*x^s*y - alpha*lambda^2*x^k + beta*lambda*x^s)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda +y} \left ( -\alpha \, \left ( \lambda +y \right ) \int \!{\frac {{x}^{k}}{{x}^{n}a+{x}^{m}b+c}{{\rm e}^{-\int \!{\frac {2\,{x}^{k}\alpha \,\lambda -{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}}\,{\rm d}x-{{\rm e}^{-\int \!{\frac {2\,{x}^{k}\alpha \,\lambda -{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.47 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(a x^n + b x^m + c) w_x - \left ( s x^k y^2 -(a x^n + b x^m+c) y - s \lambda x^{k+2} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*(a*x^n + b*x^m + c)*D[w[x, y], x] - (s*x^k*y^2 - (a*x^n + b*x^m + c)*y - s*lambda*x^(k + 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\tanh ^{-1}\left (\frac {y}{\sqrt {\lambda } x}\right )}{\sqrt {\lambda }}-\int _1^x\frac {s K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde :=x*(a*x^n + b*x^m + c)*diff(w(x,y),x)- (s*x^k*y^2 -(a*x^n + b*x^m+c)*y - s*lambda*x^(k+2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{s\sqrt {\lambda }} \left ( -\int \!{\frac {{x}^{k}}{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}xs\sqrt {\lambda }+\arctanh \left ( {\frac {y}{x\sqrt {\lambda }}} \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.48 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left ( (a x^n+b x^m + c)y^2-a n(n-1)x^{n-2}-b m(m-1) x^{m-2}\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + ((a*x^n + b*x^m + c)*y^2 - a*n*(n - 1)*x^(n - 2) - b*m*(m - 1)*x^(m - 2))*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^m + c)*diff(w(x,y),x)+ ((a*x^n+b*x^m + c)*y^2-a*n*(n-1)*x^(n-2)-b*m*(m-1)*x^(m-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( b \left ( -n+m \right ) {x}^{m}-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) \left ( {x}^{1+2\,n}y{a}^{2}+y{x}^{2\,m+1}{b}^{2}+2\,a{x}^{m+n+1}yb+ab \left ( n+m \right ) {x}^{n+m}+2\,y{x}^{m+1}bc+2\,y{x}^{n+1}ca+{x}^{2\,m}{b}^{2}m+{x}^{2\,n}{a}^{2}n+c \left ( {x}^{n}an+bm{x}^{m}+cxy \right ) \right ) } \left ( - \left ( b \left ( -n+m \right ) {x}^{m}-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) \left ( {x}^{1+2\,n}y{a}^{2}+y{x}^{2\,m+1}{b}^{2}+2\,a{x}^{m+n+1}yb+ab \left ( n+m \right ) {x}^{n+m}+2\,y{x}^{m+1}bc+2\,y{x}^{n+1}ca+{x}^{2\,m}{b}^{2}m+{x}^{2\,n}{a}^{2}n+c \left ( {x}^{n}an+bm{x}^{m}+cxy \right ) \right ) \int \!{\frac {-b \left ( n+m-1 \right ) \left ( -n+m \right ) {x}^{m}+cn \left ( n-1 \right ) }{ \left ( {x}^{n}a+{x}^{m}b+c \right ) \left ( b \left ( -n+m \right ) {x}^{m}-cn \right ) ^{2}}}\,{\rm d}x+ \left ( {x}^{1+2\,n}y{a}^{2}+y{x}^{2\,m+1}{b}^{2}+2\,a{x}^{m+n+1}yb+ab \left ( n+m \right ) {x}^{n+m}+2\,y{x}^{m+1}bc+2\,y{x}^{n+1}ca+{x}^{2\,m}{b}^{2}m+{x}^{2\,n}{a}^{2}n-{b}^{2} \left ( -n+m \right ) \left ( {x}^{m} \right ) ^{2}-b \left ( a \left ( -n+m \right ) {x}^{n}-2\,cn \right ) {x}^{m}+2\,{x}^{n}acn+{c}^{2} \left ( xy+n \right ) \right ) x \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.49 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b y^n + x) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + y \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*y^n + x)*D[w[x, y], x] + (alpha*x^k*y^(n - k) + beta*x^m*y^(n - m) + y)*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*y^n + x)*diff(w(x,y),x)+ (alpha*x^k*y^(n-k) +beta*x^m*y^(n-m) + y )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.50 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b y^n + A x^2 + B x y) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + A x y + B y^2\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*y^n + A*x^2 + B*x*y)*D[w[x, y], x] + (alpha*x^k*y^(n - k) + beta*x^m*y^(n - m) + A*x*y + B*y^2)*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*y^n + A*x^2 + B*x*y)*diff(w(x,y),x)+ (alpha*x^k*y^(n-k)+beta*x^m*y^(n-m) + A*x*y + B*y^2 )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.51 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a y^m + b x^n + s) w_x - \left ( \alpha x^k + b n x^{n-1} y + \beta \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*y^m + b*x^n + s)*D[w[x, y], x] - (alpha*x^k + b*n*x^(n - 1)*y + beta)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*y^m + b*x^n + s)*diff(w(x,y),x)- (alpha*x^k + b*n*x^(n-1)* y + beta)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-\alpha \,x \left ( m+1 \right ) {x}^{k}- \left ( k+1 \right ) \left ( by \left ( m+1 \right ) {x}^{n}+{y}^{m}ya+ \left ( \beta \,x+sy \right ) \left ( m+1 \right ) \right ) }{ \left ( k+1 \right ) \left ( m+1 \right ) }} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.52 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m +x) w_x + \left ( b x^k y^{n+m-k} + y \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + x)*D[w[x, y], x] + (b*x^k*y^(n + m - k) + y)*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 +x)*diff(w(x,y),x)+ (b*x^k*y^(n+m-k) + y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.53 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(a x^n y^m +\alpha ) w_x - y \left ( b x^n y^m + \beta \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(a*x^n*y^m + alpha)*D[w[x, y], x] - y*(b*x^n*y^m + beta)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := x*(a*x^n*y^m +alpha)*diff(w(x,y),x)- y*( b*x^n*y^m + beta )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{\beta \,m \left ( an-bm \right ) } \left ( {y}^{m} \right ) ^{\alpha \, \left ( an-bm \right ) } \left ( {y}^{m} \left ( an-bm \right ) {x}^{n}-\beta \,m+\alpha \,n \right ) ^{-m \left ( a\beta -\alpha \,b \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.54 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(a n x^k y^{n+k} + s) w_x - y \left ( b m x^{m+k} y^k + s \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(a*n*x^k*y^(n + k) + s)*D[w[x, y], x] - y*(b*m*x^(m + k)*y^k + s)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := x*(a*n*x^k*y^(n+k) + s)*diff(w(x,y),x)- y*( b*m*x^(m+k)*y^k + s )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 2, 2019.
Problem 2.2.5.55 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m + A x^2 + B x y) w_x + \left ( b x^k y^{n+m-k} + A x y+ B y^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + A*x^2 + B*x*y)*D[w[x, y], x] + (b*x^k*y^(n + m - k) + A*x*y + B*y^2)*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 + A*x^2 + B*x*y)*diff(w(x,y),x)+ (b*x^k*y^(n+m-k) + A*x*y+ B*y^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 2, 2019.
Problem 2.2.5.56 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m + b x y^k) w_x + \left ( \alpha y^s + \beta \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + b*x*y^k)*D[w[x, y], x] + (alpha*y^s + beta)*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*y^k)*diff(w(x,y),x)+ (alpha*y^s + beta)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{\frac {{y}^{m}}{\alpha \,{y}^{s}+\beta }{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac {{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}}\,{\rm d}y+{x}^{-n+1}{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac {{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}} \right ) \]
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