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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k{{\it \_a}}^{n}}{a} \left ( {\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) ^{m}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to e^{\frac {x (b-m)}{a}} \left (-\frac {c y^m x^n \left (\frac {x (b-m)}{a}\right )^{-n} \text {Gamma}\left (n+1,\frac {x (b-m)}{a}\right )}{b-m}+e^{\frac {m x}{a}} c_1\left (y e^{-\frac {x}{a}}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {c{{\it \_a}}^{n}}{a} \left ( y{{\rm e}^{{\frac {-x+{\it \_a}}{a}}}} \right ) ^{m}{{\rm e}^{-{\frac {{\it \_a}\,b}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {x}{a}}}} \right ) \right ) {{\rm e}^{{\frac {xb}{a}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to e^{a x} \left (-b y^m x^n (a x)^{-m-n} \text {Gamma}(m+n,a x)+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*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'));
\[w \left ( x,y \right ) ={\frac {b{x}^{n}{y}^{m}}{ \left ( n+m \right ) \left ( 1+n+m \right ) } \WhittakerM \left ( {\frac {n}{2}}+{\frac {m}{2}},{\frac {n}{2}}+{\frac {m}{2}}+{\frac {1}{2}},ax \right ) \left ( ax \right ) ^{-{\frac {n}{2}}-{\frac {m}{2}}}{{\rm e}^{{\frac {ax}{2}}}}}+{\frac {{x}^{n-1}{y}^{m}b}{ \left ( n+m \right ) a} \left ( ax \right ) ^{-{\frac {n}{2}}-{\frac {m}{2}}} \WhittakerM \left ( {\frac {n}{2}}+{\frac {m}{2}}+1,{\frac {n}{2}}+{\frac {m}{2}}+{\frac {1}{2}},ax \right ) {{\rm e}^{{\frac {ax}{2}}}}}+{{\rm e}^{ax}}{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\frac {1}{ \left ( n+m \right ) \left ( 1+n+m \right ) a} \left ( \left ( {\frac {a}{x}\sqrt {{x}^{2}+{y}^{2}}} \right ) ^{-n-m} \left ( {\frac {a}{x}\sqrt {{x}^{2}+{y}^{2}}} \right ) ^{n+m}{{\rm e}^{-{\frac {a}{2}\sqrt {{x}^{2}+{y}^{2}}}}} \left ( a\sqrt {{x}^{2}+{y}^{2}} \right ) ^{-{\frac {n}{2}}-{\frac {m}{2}}}b{y}^{m}{x}^{-m}{x}^{n+m} \left ( 1+n+m \right ) \WhittakerM \left ( {\frac {n}{2}}+{\frac {m}{2}}+1,{\frac {n}{2}}+{\frac {m}{2}}+{\frac {1}{2}},a\sqrt {{x}^{2}+{y}^{2}} \right ) +\sqrt {{x}^{2}+{y}^{2}}a \left ( \left ( {\frac {a}{x}\sqrt {{x}^{2}+{y}^{2}}} \right ) ^{-n-m} \left ( {\frac {a}{x}\sqrt {{x}^{2}+{y}^{2}}} \right ) ^{n+m}{{\rm e}^{-{\frac {a}{2}\sqrt {{x}^{2}+{y}^{2}}}}} \left ( a\sqrt {{x}^{2}+{y}^{2}} \right ) ^{-{\frac {n}{2}}-{\frac {m}{2}}}b{y}^{m}{x}^{-m}{x}^{n+m} \WhittakerM \left ( {\frac {n}{2}}+{\frac {m}{2}},{\frac {n}{2}}+{\frac {m}{2}}+{\frac {1}{2}},a\sqrt {{x}^{2}+{y}^{2}} \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \left ( 1+n+m \right ) \left ( n+m \right ) \right ) \right ) {{\rm e}^{a\sqrt {{x}^{2}+{y}^{2}}}}{\frac {1}{\sqrt {{x}^{2}+{y}^{2}}}}}\]
____________________________________________________________________________________
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 ( \left ( k+n \right ) a+ \left ( m+s \right ) b \right ) \left ( ak+sb \right ) c} \left ( {y}^{s} \left ( c{x}^{n}{y}^{m}+ \left ( k+n \right ) a+ \left ( m+s \right ) b \right ) \left ( na+mb \right ) ^{2}{{\rm e}^{-{\frac {c{x}^{n}{y}^{m}}{2\,na+2\,mb}}}} \left ( {\frac {{y}^{m}c}{na+mb}{x}^{-{\frac {mb}{a}}}} \right ) ^{{\frac {-ak-sb}{na+mb}}} \left ( {\frac {{y}^{m}c}{na+mb}{x}^{-{\frac {mb}{a}}}} \right ) ^{{\frac {ak+sb}{na+mb}}}{y}^{-m} \left ( {\frac {c{x}^{n}{y}^{m}}{na+mb}} \right ) ^{{\frac { \left ( -k-n \right ) a- \left ( m+s \right ) b}{2\,na+2\,mb}}}{x}^{{\frac { \left ( -n+k \right ) a+sb}{a}}}{x}^{-{\frac {sb}{a}}}p \WhittakerM \left ( {\frac { \left ( -n+k \right ) a-b \left ( m-s \right ) }{2\,na+2\,mb}},{\frac {1}{2\,na+2\,mb} \left ( \left ( k+2\,n \right ) a+2\,b \left ( m+s/2 \right ) \right ) },{\frac {c{x}^{n}{y}^{m}}{na+mb}} \right ) + \left ( {y}^{s} \left ( na+mb \right ) \left ( \left ( k+n \right ) a+ \left ( m+s \right ) b \right ) {{\rm e}^{-{\frac {c{x}^{n}{y}^{m}}{2\,na+2\,mb}}}} \left ( {\frac {{y}^{m}c}{na+mb}{x}^{-{\frac {mb}{a}}}} \right ) ^{{\frac {-ak-sb}{na+mb}}} \left ( {\frac {{y}^{m}c}{na+mb}{x}^{-{\frac {mb}{a}}}} \right ) ^{{\frac {ak+sb}{na+mb}}}{y}^{-m} \left ( {\frac {c{x}^{n}{y}^{m}}{na+mb}} \right ) ^{{\frac { \left ( -k-n \right ) a- \left ( m+s \right ) b}{2\,na+2\,mb}}}{x}^{{\frac { \left ( -n+k \right ) a+sb}{a}}}{x}^{-{\frac {sb}{a}}}p \WhittakerM \left ( {\frac { \left ( k+n \right ) a+ \left ( m+s \right ) b}{2\,na+2\,mb}},{\frac {1}{2\,na+2\,mb} \left ( \left ( k+2\,n \right ) a+2\,b \left ( m+s/2 \right ) \right ) },{\frac {c{x}^{n}{y}^{m}}{na+mb}} \right ) +{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \left ( ak+sb \right ) c \left ( \left ( k+2\,n \right ) a+2\,b \left ( m+s/2 \right ) \right ) \right ) \left ( \left ( k+n \right ) a+ \left ( m+s \right ) b \right ) \right ) {{\rm e}^{{\frac {c{x}^{n}{y}^{m}}{na+mb}}}} \left ( \left ( k+2\,n \right ) a+2\,b \left ( m+s/2 \right ) \right ) ^{-1}}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_b}}^{k-1}q}{a} \left ( y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) ^{s}{{\rm e}^{-{\frac {c}{a}\int \!{\frac {1}{{\it \_b}} \left ( {{\it \_b}}^{n}+ \left ( y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{{\it \_a}\,a} \left ( {{\it \_a}}^{n}+ \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}}}\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to e^{-\frac {b y^2}{x-2 a x}} \left (-\frac {c y^m x^{n-1} \left (-\frac {b y^2}{x-2 a x}\right )^{-\frac {a m+n-1}{2 a-1}} \text {Gamma}\left (\frac {a m+n-1}{2 a-1},-\frac {b y^2}{x-2 a x}\right )}{2 a-1}+c_1\left (y x^{-a}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =4\,{\frac {1}{xb \left ( ma+n-1 \right ) \left ( \left ( m+2 \right ) a+n-2 \right ) \left ( \left ( 4+m \right ) a+n-3 \right ) {y}^{2}}{{\rm e}^{{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}} \left ( 1/2\,x{y}^{m} \left ( {\frac {{y}^{2}{x}^{-2\,a}b}{2\,a-1}} \right ) ^{{\frac {-ma-n+1}{2\,a-1}}} \left ( a-1/2 \right ) \left ( \left ( m+2 \right ) a+n-2 \right ) ^{2} \left ( {\frac {{y}^{2}{x}^{-2\,a}b}{2\,a-1}} \right ) ^{{\frac {ma+n-1}{2\,a-1}}}c{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}{x}^{ma+n} \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{{\frac { \left ( -m-2 \right ) a-n+2}{4\,a-2}}}{x}^{-ma} \WhittakerM \left ( {\frac { \left ( m+2 \right ) a+n-2}{4\,a-2}},{\frac { \left ( 4+m \right ) a+n-3}{4\,a-2}},{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) +{y}^{m} \left ( \left ( \left ( m+2 \right ) a+n-2 \right ) x+b{y}^{2} \right ) \left ( {\frac {{y}^{2}{x}^{-2\,a}b}{2\,a-1}} \right ) ^{{\frac {-ma-n+1}{2\,a-1}}} \left ( a-1/2 \right ) ^{2} \left ( {\frac {{y}^{2}{x}^{-2\,a}b}{2\,a-1}} \right ) ^{{\frac {ma+n-1}{2\,a-1}}}c{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}{x}^{ma+n} \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{{\frac { \left ( -m-2 \right ) a-n+2}{4\,a-2}}}{x}^{-ma} \WhittakerM \left ( {\frac { \left ( m-2 \right ) a+n}{4\,a-2}},{\frac { \left ( 4+m \right ) a+n-3}{4\,a-2}},{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) +1/4\,xb{\it \_F1} \left ( y{x}^{-a} \right ) \left ( ma+n-1 \right ) \left ( \left ( m+2 \right ) a+n-2 \right ) \left ( \left ( 4+m \right ) a+n-3 \right ) {y}^{2} \right ) }\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to -c 2^{\frac {1}{2} (m+n-3)} y^m x^{n-1} e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )} \left (y^2 \left (a+\frac {b y}{x}\right )\right )^{\frac {1}{2} (-m-n+1)} \text {Gamma}\left (\frac {1}{2} (m+n-1),\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )\right )+c_1\left (\frac {y}{x}\right ) e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )}\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\frac {1}{{y}^{2} \left ( n+m+3 \right ) \left ( n-1+m \right ) \left ( 1+n+m \right ) {x}^{2}}{{\rm e}^{{\frac {b{y}^{3}}{2\,x}}+{\frac {a{y}^{2}}{2}}}} \left ( x{2}^{{\frac {n}{2}}+{\frac {m}{2}}+{\frac {1}{2}}}{y}^{m+2} \left ( \left ( a{y}^{2}+m+n+1 \right ) x+b{y}^{3} \right ) {x}^{-m-2} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-{\frac {n}{2}}-{\frac {m}{2}}-{\frac {1}{2}}} \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) ^{-{\frac {n}{4}}-{\frac {m}{4}}-{\frac {1}{4}}}{{\rm e}^{-{\frac {b{y}^{3}}{4\,x}}-{\frac {a{y}^{2}}{4}}}}{x}^{n-1+m}c \left ( n+m+3 \right ) {2}^{-{\frac {n}{4}}-{\frac {m}{4}}+{\frac {3}{4}}} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-{\frac {1}{2}}+{\frac {n}{2}}+{\frac {m}{2}}} \WhittakerM \left ( {\frac {n}{4}}+{\frac {m}{4}}+{\frac {5}{4}},{\frac {n}{4}}+{\frac {m}{4}}+{\frac {3}{4}},{\frac {{y}^{2}}{2} \left ( {\frac {by}{x}}+a \right ) } \right ) + \left ( {x}^{n-1+m}{2}^{-{\frac {n}{4}}-{\frac {m}{4}}+{\frac {3}{4}}}{2}^{{\frac {n}{2}}+{\frac {m}{2}}+{\frac {1}{2}}}{{\rm e}^{-{\frac {b{y}^{3}}{4\,x}}-{\frac {a{y}^{2}}{4}}}} \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) ^{-{\frac {n}{4}}-{\frac {m}{4}}-{\frac {1}{4}}} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-{\frac {1}{2}}+{\frac {n}{2}}+{\frac {m}{2}}} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-{\frac {n}{2}}-{\frac {m}{2}}-{\frac {1}{2}}}c{y}^{2}{x}^{-m-2}{y}^{m+2} \left ( ax+by \right ) ^{2} \WhittakerM \left ( {\frac {n}{4}}+{\frac {m}{4}}+{\frac {1}{4}},{\frac {n}{4}}+{\frac {m}{4}}+{\frac {3}{4}},{\frac {{y}^{2}}{2} \left ( {\frac {by}{x}}+a \right ) } \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) {x}^{2} \left ( n-1+m \right ) \left ( n+m+3 \right ) \left ( 1+n+m \right ) \right ) {y}^{2} \right ) }\]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {c}{a}\int \!{{\it \_b}}^{-n+p} \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{m-n+1}-{{\it \_b}}^{m-n+1} \right ) }{ \left ( m-n+1 \right ) a}}}} \right ) ^{q}\,{\rm d}{\it \_b}}}} \left ( {{\it \_b}}^{-n+\gamma }s \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{m-n+1}-{{\it \_b}}^{m-n+1} \right ) }{ \left ( m-n+1 \right ) a}}}} \right ) ^{\delta }+{{\it \_b}}^{-n}d \right ) }{d{\it \_b}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-n+p}c}{a} \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{m-n+1}-{{\it \_a}}^{m-n+1} \right ) }{ \left ( m-n+1 \right ) a}}}} \right ) ^{q}}{d{\it \_a}}}}\]
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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 ( 1-n+k \right ) } \left ( a{{\it \_b}}^{-m+k}{{\rm e}^{{\frac {{{\it \_b}}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( {\frac {{{\it \_b}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{{\it \_b}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -a{x}^{-m+k}{{\rm e}^{{\frac {{{\it \_b}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}}{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( {\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) + \left ( m-n+1 \right ) ^{2}c \left ( b{{\it \_b}}^{1-n+k}+a{{\it \_b}}^{-m+k} \left ( k+m-2\,n+2 \right ) \right ) \left ( {\frac {{{\it \_b}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}{{\rm e}^{{\frac {{{\it \_b}}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \WhittakerM \left ( {\frac {-m+k}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{{\it \_b}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -{{\rm e}^{{\frac {{{\it \_b}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}} \left ( {{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( b{x}^{1-n+k}+a{x}^{-m+k} \left ( k+m-2\,n+2 \right ) \right ) \left ( m-n+1 \right ) ^{2} \left ( {\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \WhittakerM \left ( {\frac {-m+k}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}}yab \left ( 1-n+k \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \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 ( 1-n+k \right ) } \left ( -a{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( {\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c{x}^{-m+k} \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( b{x}^{1-n+k}+a{x}^{-m+k} \left ( k+m-2\,n+2 \right ) \right ) \left ( m-n+1 \right ) ^{2} \left ( {\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \WhittakerM \left ( {\frac {-m+k}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) +{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}}yab \left ( 1-n+k \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \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 ( 1-n+k \right ) } \left ( a{{\it \_a}}^{-m+k}{{\rm e}^{{\frac {{{\it \_a}}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( {\frac {{{\it \_a}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{{\it \_a}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -a{x}^{-m+k}{{\rm e}^{{\frac {{{\it \_a}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}}{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( {\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \left ( m-n+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) + \left ( m-n+1 \right ) ^{2}c \left ( b{{\it \_a}}^{1-n+k}+a{{\it \_a}}^{-m+k} \left ( k+m-2\,n+2 \right ) \right ) \left ( {\frac {{{\it \_a}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}{{\rm e}^{{\frac {{{\it \_a}}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \WhittakerM \left ( {\frac {-m+k}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{{\it \_a}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -{{\rm e}^{{\frac {{{\it \_a}}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}} \left ( {{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( 2\,m-2\,n+2 \right ) a}}}} \left ( b{x}^{1-n+k}+a{x}^{-m+k} \left ( k+m-2\,n+2 \right ) \right ) \left ( m-n+1 \right ) ^{2} \left ( {\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) ^{{\frac {-k-m+2\,n-2}{2\,m-2\,n+2}}}c \WhittakerM \left ( {\frac {-m+k}{2\,m-2\,n+2}},{\frac {k+2\,m-3\,n+3}{2\,m-2\,n+2}},{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}} \right ) -{{\rm e}^{-{\frac {{x}^{m-n+1}b}{ \left ( m-n+1 \right ) a}}}}yab \left ( 1-n+k \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {{{\it \_a}}^{1-n}c}{ \left ( n-1 \right ) a}}}} \left ( {{\it \_a}}^{-n+p}s \left ( \left ( {\frac {{x}^{m-n+1} \left ( k-1 \right ) b-b \left ( k-1 \right ) {{\it \_a}}^{m-n+1}+a{y}^{-k+1} \left ( m-n+1 \right ) }{ \left ( m-n+1 \right ) a}} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {{x}^{m-n+1} \left ( k-1 \right ) b+a{y}^{-k+1} \left ( m-n+1 \right ) }{ \left ( m-n+1 \right ) a}} \right ) \right ) {{\rm e}^{-{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {s{{\it \_b}}^{m}}{a} \left ( \left ( {\frac {-{x}^{1+n}b \left ( 1+k \right ) +{y}^{1+k}a \left ( 1+n \right ) +{{\it \_b}}^{1+n}b \left ( 1+k \right ) }{ \left ( 1+n \right ) a}} \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \left ( {\frac {-{x}^{1+n}b \left ( 1+k \right ) +{y}^{1+k}a \left ( 1+n \right ) +{{\it \_b}}^{1+n}b \left ( 1+k \right ) }{ \left ( 1+n \right ) a}} \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac {-{x}^{1+n}b \left ( 1+k \right ) +{y}^{1+k}a \left ( 1+n \right ) }{ \left ( 1+n \right ) a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( \left ( {\frac {-{x}^{1+n}b \left ( 1+k \right ) +{y}^{1+k}a \left ( 1+n \right ) +{{\it \_a}}^{1+n}b \left ( 1+k \right ) }{ \left ( 1+n \right ) a}} \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}}\]
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