Added March 12, 2019.
Problem Chapter 5.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + k x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + k*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} k K[1]^n \left (y+\frac {b (K[1]-x)}{a}\right )^m}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ k*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k{{\it \_a}}^{n}}{a} \left ( {\frac {ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) ^{m}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + y w_y = b w + c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + y*D[w[x, y], y] == b*w[x, y] + c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x (b-m)}{a}} \left (e^{\frac {m x}{a}} c_1\left (y e^{-\frac {x}{a}}\right )-\frac {c y^m x^n \left (\frac {x (b-m)}{a}\right )^{-n} \text {Gamma}\left (n+1,\frac {x (b-m)}{a}\right )}{b-m}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ y*diff(w(x,y),y) = b*w(x,y)+ c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {c{{\it \_a}}^{n}}{a} \left ( y{{\rm e}^{{\frac {{\it \_a}-x}{a}}}} \right ) ^{m}{{\rm e}^{-{\frac {b{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {x}{a}}}} \right ) \right ) {{\rm e}^{{\frac {bx}{a}}}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x w + b x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*w[x, y] + b*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{a x} \left (c_1\left (\frac {y}{x}\right )-b y^m x^n (a x)^{-m-n} \text {Gamma}(m+n,a x)\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*w(x,y)+ b*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {{x}^{n}{y}^{m} \WhittakerM \left ( n/2+m/2,m/2+n/2+1/2,ax \right ) b \left ( ax \right ) ^{-n/2-m/2}{{\rm e}^{1/2\,ax}}}{ \left ( n+m \right ) \left ( m+n+1 \right ) }}+{\frac {{x}^{n-1}{y}^{m} \WhittakerM \left ( n/2+m/2+1,m/2+n/2+1/2,ax \right ) b \left ( ax \right ) ^{-n/2-m/2}{{\rm e}^{1/2\,ax}}}{ \left ( n+m \right ) a}}+{{\rm e}^{ax}}{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a \sqrt {x^2+y^2} w + b x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*Sqrt[x^2+y^2]*w[x,y] + b*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{a \sqrt {x^2+y^2}} \left (\int _1^xb e^{-a \sqrt {\left (\frac {y^2}{x^2}+1\right ) K[1]^2}} K[1]^{n-1} \left (\frac {y K[1]}{x}\right )^mdK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*sqrt(x^2+y^2)*w(x,y)+ b*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {{{\rm e}^{a\sqrt {{x}^{2}+{y}^{2}}}}}{\sqrt {{x}^{2}+{y}^{2}} \left ( n+m \right ) \left ( m+n+1 \right ) a} \left ( \left ( {\frac {a\sqrt {{x}^{2}+{y}^{2}}}{x}} \right ) ^{-n-m} \left ( {\frac {a\sqrt {{x}^{2}+{y}^{2}}}{x}} \right ) ^{n+m}{{\rm e}^{-1/2\,a\sqrt {{x}^{2}+{y}^{2}}}} \left ( a\sqrt {{x}^{2}+{y}^{2}} \right ) ^{-n/2-m/2}b{y}^{m}{x}^{-m}{x}^{n+m} \left ( m+n+1 \right ) \WhittakerM \left ( n/2+m/2+1,m/2+n/2+1/2,a\sqrt {{x}^{2}+{y}^{2}} \right ) +\sqrt {{x}^{2}+{y}^{2}} \left ( \left ( {\frac {a\sqrt {{x}^{2}+{y}^{2}}}{x}} \right ) ^{-n-m} \left ( {\frac {a\sqrt {{x}^{2}+{y}^{2}}}{x}} \right ) ^{n+m}{{\rm e}^{-1/2\,a\sqrt {{x}^{2}+{y}^{2}}}} \left ( a\sqrt {{x}^{2}+{y}^{2}} \right ) ^{-n/2-m/2}b{y}^{m}{x}^{-m}{x}^{n+m} \WhittakerM \left ( n/2+m/2,m/2+n/2+1/2,a\sqrt {{x}^{2}+{y}^{2}} \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \left ( m+n+1 \right ) \left ( n+m \right ) \right ) a \right ) }\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c x^n y^m w + p x^k y^s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n*y^m*w[x,y] + p*x^k*y^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c y^m x^n}{a n+b m}} \left (\int _1^x\frac {\exp \left (-\frac {c K[1]^n \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^m}{b m+a n}\right ) p K[1]^{k-1} \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^s}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*x^n*y^m*w(x,y)+ p*x^k*y^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{ \left ( ak+sb \right ) c \left ( a \left ( 2\,n+k \right ) +2\, \left ( m+s/2 \right ) b \right ) \left ( \left ( n+k \right ) a+ \left ( m+s \right ) b \right ) } \left ( \left ( an+bm \right ) ^{2} \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{{\frac {ak+sb}{an+bm}}} \left ( c{x}^{n}{y}^{m}+ \left ( n+k \right ) a+ \left ( m+s \right ) b \right ) \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{{\frac {-ak-sb}{an+bm}}} \left ( {\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) ^{{\frac { \left ( -k-n \right ) a- \left ( m+s \right ) b}{2\,an+2\,bm}}}{{\rm e}^{-{\frac {c{x}^{n}{y}^{m}}{2\,an+2\,bm}}}}p{y}^{-m}{x}^{{\frac { \left ( k-n \right ) a+sb}{a}}}{x}^{-{\frac {sb}{a}}}{y}^{s} \WhittakerM \left ( {\frac { \left ( k-n \right ) a-b \left ( m-s \right ) }{2\,an+2\,bm}},{\frac {a \left ( 2\,n+k \right ) +2\, \left ( m+s/2 \right ) b}{2\,an+2\,bm}},{\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) + \left ( \left ( n+k \right ) a+ \left ( m+s \right ) b \right ) \left ( \left ( an+bm \right ) \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{{\frac {ak+sb}{an+bm}}} \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{{\frac {-ak-sb}{an+bm}}} \left ( {\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) ^{{\frac { \left ( -k-n \right ) a- \left ( m+s \right ) b}{2\,an+2\,bm}}}{{\rm e}^{-{\frac {c{x}^{n}{y}^{m}}{2\,an+2\,bm}}}}p{y}^{-m}{x}^{{\frac { \left ( k-n \right ) a+sb}{a}}} \left ( \left ( n+k \right ) a+ \left ( m+s \right ) b \right ) {x}^{-{\frac {sb}{a}}}{y}^{s} \WhittakerM \left ( {\frac { \left ( n+k \right ) a+ \left ( m+s \right ) b}{2\,an+2\,bm}},{\frac {a \left ( 2\,n+k \right ) +2\, \left ( m+s/2 \right ) b}{2\,an+2\,bm}},{\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) + \left ( ak+sb \right ) {\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) c \left ( a \left ( 2\,n+k \right ) +2\, \left ( m+s/2 \right ) b \right ) \right ) \right ) {{\rm e}^{{\frac {c{x}^{n}{y}^{m}}{an+bm}}}}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = (c x^n+ p y^m) w + q x^k y^s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*(x^n+p*y^m)*w[x,y] + q*x^k*y^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x^n}{a n}+\frac {c p y^m}{b m}} \left (\int _1^x\frac {\exp \left (-\frac {c \left (\frac {a p \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^m}{b m}+\frac {K[1]^n}{n}\right )}{a}\right ) q K[1]^{k-1} \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^s}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*(x^n+y^m)*w(x,y)+ q*x^k*y^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {q{{\it \_b}}^{k-1}}{a} \left ( y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) ^{s}{{\rm e}^{-{\frac {c}{a}\int \!{\frac {1}{{\it \_b}} \left ( {{\it \_b}}^{n}+ \left ( y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{{\it \_a}\,a} \left ( {{\it \_a}}^{n}+ \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2 w_x + a x y w_y = b y^2 w + c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2*w[x,y] + c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {b y^2}{x-2 a x}} \left (c_1\left (y x^{-a}\right )-\frac {c y^m x^{n-1} \left (-\frac {b y^2}{x-2 a x}\right )^{-\frac {a m+n-1}{2 a-1}} \text {Gamma}\left (\frac {a m+n-1}{2 a-1},-\frac {b y^2}{x-2 a x}\right )}{2 a-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := x^2*diff(w(x,y),x)+ a*x*y*diff(w(x,y),y) = b*y^2*w(x,y)+ c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =4\,{\frac {1}{ \left ( a \left ( m+2 \right ) +n-2 \right ) \left ( ma+n-1 \right ) b \left ( a \left ( m+4 \right ) +n-3 \right ) {y}^{2}x} \left ( 1/2\, \left ( a \left ( m+2 \right ) +n-2 \right ) ^{2}{y}^{m}{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}{x}^{ma+n} \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {-ma-n+1}{2\,a-1}}} \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {ma+n-1}{2\,a-1}}}c{x}^{-ma}x \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{{\frac { \left ( -m-2 \right ) a-n+2}{4\,a-2}}} \left ( a-1/2 \right ) \WhittakerM \left ( {\frac {a \left ( m+2 \right ) +n-2}{4\,a-2}},{\frac {a \left ( m+4 \right ) +n-3}{4\,a-2}},{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) +{y}^{m}{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}{x}^{ma+n} \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {-ma-n+1}{2\,a-1}}} \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {ma+n-1}{2\,a-1}}} \left ( \left ( a \left ( m+2 \right ) +n-2 \right ) x+b{y}^{2} \right ) c{x}^{-ma} \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{{\frac { \left ( -m-2 \right ) a-n+2}{4\,a-2}}} \left ( a-1/2 \right ) ^{2} \WhittakerM \left ( {\frac { \left ( -2+m \right ) a+n}{4\,a-2}},{\frac {a \left ( m+4 \right ) +n-3}{4\,a-2}},{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) +1/4\, \left ( a \left ( m+2 \right ) +n-2 \right ) \left ( ma+n-1 \right ) b \left ( a \left ( m+4 \right ) +n-3 \right ) {y}^{2}x{\it \_F1} \left ( y{x}^{-a} \right ) \right ) {{\rm e}^{{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2 w_x + x y w_y = y^2(a x+b y) w + c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x^2*D[w[x, y], x] + x*y*D[w[x, y], y] == y^2*(a*x+b*y)*w[x,y] + c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )}-c 2^{\frac {1}{2} (m+n-3)} y^m x^{n-1} e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )} \left (y^2 \left (a+\frac {b y}{x}\right )\right )^{\frac {1}{2} (-m-n+1)} \text {Gamma}\left (\frac {1}{2} (m+n-1),\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x^2*diff(w(x,y),x)+ x*y*diff(w(x,y),y) = y^2*(a*x+b*y)*w(x,y)+ c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{{y}^{2} \left ( m+n+1 \right ) {x}^{2} \left ( n+m-1 \right ) \left ( n+3+m \right ) }{{\rm e}^{1/2\,{\frac {b{y}^{3}}{x}}+1/2\,{y}^{2}a}} \left ( {x}^{-m-2}{x}^{n+m-1}{y}^{m+2}{2}^{m/2+n/2+1/2} \left ( \left ( {y}^{2}a+m+n+1 \right ) x+b{y}^{3} \right ) \left ( n+3+m \right ) \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-m/2-n/2-1/2} \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) ^{-m/4-n/4-1/4}{{\rm e}^{-1/4\,{\frac {b{y}^{3}}{x}}-1/4\,{y}^{2}a}}cx \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-1/2+n/2+m/2}{2}^{-n/4-m/4+3/4} \WhittakerM \left ( m/4+n/4+5/4,n/4+m/4+3/4,1/2\,{y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) +{y}^{2} \left ( {x}^{n+m-1}{2}^{m/2+n/2+1/2}{2}^{-n/4-m/4+3/4}{{\rm e}^{-1/4\,{\frac {b{y}^{3}}{x}}-1/4\,{y}^{2}a}} \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) ^{-m/4-n/4-1/4} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-1/2+n/2+m/2} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-m/2-n/2-1/2}c{y}^{2}{x}^{-m-2}{y}^{m+2} \left ( ax+by \right ) ^{2} \WhittakerM \left ( m/4+n/4+1/4,n/4+m/4+3/4,1/2\,{y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) {x}^{2} \left ( m+n+1 \right ) \left ( n+m-1 \right ) \left ( n+3+m \right ) \right ) \right ) }\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b x^m y w_y = c x^p y^q w+s x^\gamma y^\delta + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == c*x^p*y^q*w[x,y] + s*x^gamma*y^delta+d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (-\frac {c y^q x^{-n+p+1} e^{-\frac {b q x^{m-n+1}}{a m-a n+a}} \left (-\frac {b q x^{m-n+1}}{a m-a n+a}\right )^{\frac {n-p-1}{m-n+1}} \text {Gamma}\left (\frac {-n+p+1}{m-n+1},-\frac {b q x^{m-n+1}}{a m-a n+a}\right )}{a (m-n+1)}\right ) \left (\int _1^x\frac {\exp \left (\frac {c e^{-\frac {b q K[1]^{m-n+1}}{m a-n a+a}} \left (e^{\frac {b \left (K[1]^{m-n+1}-x^{m-n+1}\right )}{m a-n a+a}} y\right )^q \text {Gamma}\left (\frac {-n+p+1}{m-n+1},-\frac {b q K[1]^{m-n+1}}{m a-n a+a}\right ) K[1]^{-n+p+1} \left (-\frac {b q K[1]^{m-n+1}}{m a-n a+a}\right )^{\frac {n-p-1}{m-n+1}}}{a (m-n+1)}\right ) K[1]^{-n} \left (s K[1]^{\gamma } \left (e^{\frac {b \left (K[1]^{m-n+1}-x^{m-n+1}\right )}{m a-n a+a}} y\right )^{\delta }+d\right )}{a}dK[1]+c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y),x)+ b*x^m*y*diff(w(x,y),y) = c*x^p*y^q*w(x,y)+ s*x^gamma*y^delta+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{-n+m+1}-{{\it \_b}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{q}{{\it \_b}}^{-n+p}\,{\rm d}{\it \_b}}}} \left ( {{\it \_b}}^{-n+1/8}s \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{-n+m+1}-{{\it \_b}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{\delta }+{{\it \_b}}^{-n}d \right ) }{d{\it \_b}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-n+p}c}{a} \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{-n+m+1}-{{\it \_a}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{q}}{d{\it \_a}}}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + (b x^m y +c x^k ) w_y = s x^p y^q w+d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + (b*x^m*y+x*x^k)*D[w[x, y], y] == s*x^p*y^q*w[x,y] + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {s \left (b^{-\frac {k+2}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}-b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}+b^{\frac {k+2}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^{p-n}}{a}dK[1]\right ) \left (\int _1^x\frac {d \exp \left (-\int _1^{K[2]}\frac {s \left (b^{-\frac {k+2}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}-b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}+b^{\frac {k+2}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^{p-n}}{a}dK[1]\right ) K[2]^{-n}}{a}dK[2]+c_1\left ((a (m-n+1))^{\frac {k-m+1}{m-n+1}} b^{\frac {-k+n-2}{m-n+1}} \text {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y),x)+ (b*x^m*y+c*x^k)*diff(w(x,y),y) = s*x^p*y^q*w(x,y)+ d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_b}}^{-n}d}{a}{{\rm e}^{-{\frac {s}{a}\int \! \left ( {\frac {1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( a{{\it \_b}}^{k-m}{{\rm e}^{1/2\,{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -a{x}^{k-m}{{\rm e}^{{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) + \left ( -n+m+1 \right ) ^{2} \left ( {\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( b{{\it \_b}}^{k-n+1}+a{{\it \_b}}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) {{\rm e}^{1/2\,{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ( \left ( -n+m+1 \right ) ^{2} \left ( b{x}^{k-n+1}+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+m-2\,n+2 \right ) \left ( k+2\,m-3\,n+3 \right ) \right ) {{\rm e}^{{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) } \right ) ^{q}{{\it \_b}}^{-n+p}\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( -a{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{x}^{k-m} \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ( -n+m+1 \right ) ^{2} \left ( b{x}^{k-n+1}+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) +{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+m-2\,n+2 \right ) \left ( k+2\,m-3\,n+3 \right ) \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-n+p}s}{a} \left ( {\frac {1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( a{{\it \_a}}^{k-m}{{\rm e}^{1/2\,{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -a{x}^{k-m}{{\rm e}^{{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) + \left ( -n+m+1 \right ) ^{2} \left ( {\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( b{{\it \_a}}^{k-n+1}+a{{\it \_a}}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) {{\rm e}^{1/2\,{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ( \left ( -n+m+1 \right ) ^{2} \left ( b{x}^{k-n+1}+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+m-2\,n+2 \right ) \left ( k+2\,m-3\,n+3 \right ) \right ) {{\rm e}^{{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) } \right ) ^{q}}{d{\it \_a}}}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b x^m y^k w_y = c w + s x^p y^q + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y^k*D[w[x, y], y] == c*w[x,y] + s*x^p*y^q+d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x^{1-n}}{a-a n}} \left (\int _1^x\frac {e^{\frac {c K[1]^{1-n}}{a (n-1)}} K[1]^{-n} \left (s \left (\left (-\frac {a (m-n+1) x^n y^k K[1]^n}{a (-m+n-1) x^n y K[1]^n-b (k-1) y^k \left (x^{m+1} K[1]^n-x^n K[1]^{m+1}\right )}\right )^{\frac {1}{k-1}}\right )^q K[1]^p+d\right )}{a}dK[1]+c_1\left (\frac {\frac {y^{1-k} (-m+n-1)}{k-1}-\frac {b x^{m-n+1}}{a}}{m-n+1}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y),x)+ b*x^m*y^k*diff(w(x,y),y) = c*w(x,y)+ s*x^p*y^q+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {{{\it \_a}}^{-n+1}c}{a \left ( n-1 \right ) }}}} \left ( {{\it \_a}}^{-n+p}s \left ( \left ( {\frac {{x}^{-n+m+1}b \left ( k-1 \right ) -b \left ( k-1 \right ) {{\it \_a}}^{-n+m+1}+{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {{x}^{-n+m+1}b \left ( k-1 \right ) +{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) \right ) {{\rm e}^{-{\frac {{x}^{-n+1}c}{a \left ( n-1 \right ) }}}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^k w_x + b x^n w_y = c w + s x^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == c*w[x,y] + s*x^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right ) K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}\right ) s K[1]^m \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*y^k*diff(w(x,y),x)+ b*x^n*diff(w(x,y),y) = c*w(x,y)+ s*x^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {s{{\it \_b}}^{m}}{a} \left ( \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) +b{{\it \_b}}^{n+1} \left ( k+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) +b{{\it \_b}}^{n+1} \left ( k+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) +b{{\it \_a}}^{n+1} \left ( k+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}}\]
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