Added Feb. 17, 2019.
Problem Chapter 4.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = (c x^n + d y^m) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*x^n + d*y^m)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {c x^{n+1}}{a n+a}+\frac {d y^{m+1}}{b m+b}}\right \}\right \}\]
Maple ✓
restart; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = (c*x^n + d*y^m)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (n +1\right ) a d \,y^{m +1}+\left (m +1\right ) b c \,x^{n +1}}{\left (n +1\right ) \left (m +1\right ) a b}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.2 case \(n eq -1, n eq -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c x^n y w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol[[2]] = Assuming[{n != -1, n != -2}, Simplify[sol[[2]]]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c x^{n+1} (a (n+2) y-b x)}{a^2 (n+1) (n+2)}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^n*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol) assuming n<>-1, n<>-2;
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (\left (n +2\right ) a y \,x^{n +1}-b \,x^{n +2}\right ) c}{\left (n +2\right ) \left (n +1\right ) a^{2}}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.2 case \(n= -1\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c x^n y w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> n == -1], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {c (\log (x) (a y-b x)+b x)}{a^2}}\right \}\right \}\]
Maple ✓
restart; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^n*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n=-1),output='realtime'));
\[w \left (x , y\right ) = x^{\frac {\left (a y -b x \right ) c}{a^{2}}} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {b c x}{a^{2}}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.2 case \(n= -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c x^n y w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> n == -2], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) e^{\frac {c (-a y+b x+b x \log (x))}{a^2 x}}\right \}\right \}\]
Maple ✓
restart; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^n*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n=-2),output='realtime'));
\[w \left (x , y\right ) = x^{\frac {b c}{a^{2}}} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{-\frac {\left (a y -b x \right ) c}{a^{2} x}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + y w_y = a(x^2+y^2)^k w \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*(x^2 + y^2)^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {a \left (x^2+y^2\right )^k}{2 k}}\right \}\right \}\]
Maple ✓
restart; pde :=x*diff(w(x,y),x)+y*diff(w(x,y),y) = a*(x^2+y^2)^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{\frac {a \left (x^{2}+y^{2}\right )^{k}}{2 k}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x w_x + b y w_y = c x^n y^m w \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right ) e^{\frac {c y^m x^n}{a n+b m}}\right \}\right \}\]
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right ) {\mathrm e}^{\frac {c \,x^{n} y^{m}}{n a +b m}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x w_x + b y w_y = (c x^n + k y^m) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == (c*x^n + k*y^m)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right ) e^{\frac {c x^n}{a n}+\frac {k y^m}{b m}}\right \}\right \}\]
Maple ✓
restart; pde :=a*x*diff(w(x,y),x)+b*y*diff(w(x,y),y) = (c*x^n + k*y^m)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right ) {\mathrm e}^{\frac {a k n \,y^{m}+b c m \,x^{n}}{a b m n}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ m x w_x + n y w_y = (a x^n + b y^m)^k w \]
Mathematica ✓
ClearAll["Global`*"]; pde = m*x*D[w[x, y], x] + n*y*D[w[x, y], y] == (a*x^n + b*y^m)^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {n}{m}}\right ) e^{\frac {\left (a x^n+b y^m\right )^k}{k m n}}\right \}\right \}\]
Maple ✓
restart; pde :=m*x*diff(w(x,y),x)+n*y*diff(w(x,y),y) = (a*x^n + b*y^m)^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,x^{-\frac {n}{m}}\right ) {\mathrm e}^{\frac {\left (a \,x^{n}+b \,y^{m}\right )^{k}}{k m n}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x + b y^m w_y = (c x^k + d y^s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*y^m*D[w[x, y], y] == (c*x^k + d*y^s)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {y^{1-m}}{m-1}\right ) \exp \left (\frac {\frac {c x^{k-n+1}}{a}+\frac {d (-k+n-1) y^{1-m} \left (\left (y^{m-1}\right )^{\frac {1}{m-1}}\right )^s}{b (m-s-1)}}{k-n+1}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x)+b*y^m*diff(w(x,y),y) = (c*x^k + d*y^s)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (n -1\right ) a \,y^{-m +1}-\left (m -1\right ) b \,x^{-n +1}}{\left (n -1\right ) a}\right ) {\mathrm e}^{\frac {-\left (k -n +1\right ) a^{2} d \,a^{\frac {s}{m -1}-1} y^{-m +1} \left (\left (n -1\right ) a \,y^{-m +1}\right )^{-\frac {s}{m -1}} \left (n -1\right )^{\frac {s}{m -1}} {\mathrm e}^{-\frac {i \left (\mathrm {csgn}\left (\frac {i}{n -1}\right ) \mathrm {csgn}\left (i a \,y^{-m +1}\right )^{2}-\mathrm {csgn}\left (i a \,y^{-m +1}\right )^{3}-\left (-\mathrm {csgn}\left (\frac {i}{a}\right )+\mathrm {csgn}\left (i y^{-m +1}\right )\right ) \mathrm {csgn}\left (i y^{-m +1}\right )^{2}+\left (-\mathrm {csgn}\left (\frac {i}{a}\right )+\mathrm {csgn}\left (i y^{-m +1}\right )\right ) \mathrm {csgn}\left (i y^{-m +1}\right ) \mathrm {csgn}\left (i a \,y^{-m +1}\right )+\left (-\mathrm {csgn}\left (\frac {i}{n -1}\right )+\mathrm {csgn}\left (i a \,y^{-m +1}\right )\right ) \mathrm {csgn}\left (i a \,y^{-m +1}\right ) \mathrm {csgn}\left (i \left (n -1\right ) a \,y^{-m +1}\right )\right ) \pi s}{2 m -2}}+\left (m -s -1\right ) b c \,x^{k -n +1}}{\left (k -n +1\right ) \left (m -s -1\right ) a b}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x + b x^m y w_y = (c x^k y^s + d) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == (c*x^k*y^s + d)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\frac {x^{1-n} \left (\frac {d}{1-n}-\frac {c x^k y^s e^{-\frac {b s x^{m-n+1}}{a m-a n+a}} \left (-\frac {b s x^{m-n+1}}{a m-a n+a}\right )^{\frac {-k+n-1}{m-n+1}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a m-a n+a}\right )}{m-n+1}\right )}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x)+b*x^m*y*diff(w(x,y),y) = (c*x^k*y^s + d)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (c \,\textit {\_a}^{k} \left (y \,{\mathrm e}^{-\frac {\left (-\textit {\_a}^{m -n +1}+x^{m -n +1}\right ) b}{\left (m -n +1\right ) a}}\right )^{s}+d \right ) \textit {\_a}^{-n}}{a}d \textit {\_a}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x + (b x^m y+c x^k) w_y = (s x^p y^q + d) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + (b*x^m*y + c*x^k)*D[w[x, y], y] == (s*x^p*y^q + d)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (c (a (m-n+1))^{\frac {k-m}{m-n+1}} b^{\frac {-k+n-1}{m-n+1}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\int _1^x\frac {K[1]^{-n} \left (s \left (b^{-\frac {k+1}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}-b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}+b^{\frac {k+1}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^p+d\right )}{a}dK[1]\right )\right \}\right \}\]
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 + d)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,x^{k -m} \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +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 {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (k -n +1\right ) \left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) a b y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}-\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,x^{k -m}+b \,x^{k -n +1}\right ) c \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}}{\left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) \left (k -n +1\right ) a b}\right ) {\mathrm e}^{\int _{}^{x}\frac {s \,\textit {\_a}^{-n +p} \left (\frac {-\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,x^{k -m} \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +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 {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}} {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,\textit {\_a}^{k -m} \left (\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +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 {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,\textit {\_a}^{k -m}+b \,\textit {\_a}^{k -n +1}\right ) c \left (\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{2 \left (m -n +1\right ) a}}-\left (-\left (k -n +1\right ) \left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) a b y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}+\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,x^{k -m}+b \,x^{k -n +1}\right ) c \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}}}{\left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) \left (k -n +1\right ) a b}\right )^{q}+d \,\textit {\_a}^{-n}}{a}d \textit {\_a}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x + b x^m y^k w_y = (c x^p y^q + s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y^k*D[w[x, y], y] == (c*x^p*y^q + s)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\frac {y^{1-k} (-m+n-1)}{k-1}-\frac {b x^{m-n+1}}{a}}{m-n+1}\right ) \exp \left (\frac {x^{1-n} \left (-\frac {c x^p \left (\left (y^{k-1}\right )^{\frac {1}{k-1}}\right )^q \left (\frac {a y (m-n+1) x^n}{a y (m-n+1) x^n+b (k-1) y^k x^{m+1}}\right )^{\frac {q}{k-1}} \, _2F_1\left (\frac {-n+p+1}{m-n+1},\frac {q}{k-1};\frac {m-2 n+p+2}{m-n+1};\frac {b (k-1) x^{m+1} y^k}{b (k-1) x^{m+1} y^k+a (m-n+1) x^n y}\right )}{n-p-1}-\frac {s}{n-1}\right )}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x)+b*x^m*y^k*diff(w(x,y),y) = (c*x^p*y^q + s)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (m -n +1\right ) a \,y^{-k +1}+\left (k -1\right ) b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (c \,\textit {\_a}^{p} \left (\left (\frac {\left (m -n +1\right ) a \,y^{-k +1}-\left (k -1\right ) b \,\textit {\_a}^{m -n +1}+\left (k -1\right ) b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{-\frac {1}{k -1}}\right )^{q}+s \right ) \textit {\_a}^{-n}}{a}d \textit {\_a}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a y^k w_x + b x^n w_y = (c x^m + s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == (c*x^m + s)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right ) \exp \left (\frac {x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \left (c x^m \, _2F_1\left (\frac {k}{k+1},\frac {m+1}{n+1};\frac {m+n+2}{n+1};\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )+(m+1) s \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )\right )}{a (m+1)}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*y^k*diff(w(x,y),x)+b*x^n*diff(w(x,y),y) = (c*x^m+ s)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (n +1\right ) a \,y^{k +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (c \,\textit {\_a}^{m}+s \right ) \left (\left (\frac {\left (n +1\right ) a \,y^{k +1}+\left (k +1\right ) b \,\textit {\_a}^{n +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {1}{k +1}}\right )^{-k}}{a}d \textit {\_a}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x(x^n + (a n -1) y^n) w_x + y(y^n + (a n -1) x^n) w_y = k n (x^n + y^n) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*(x^n + (a*n - 1)*y^n)*D[w[x, y], x] + y*(y^n + (a*n - 1)*x^n)*D[w[x, y], y] == k*n*(x^n + y^n)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := x*(x^n + (a*n -1)*y^n)*diff(w(x,y),x)+y*(y^n + (a*n -1)*x^n)*diff(w(x,y),y) = k*n*(x^n + y^n)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (-x^{-\frac {1}{a}} y^{n}+x^{\frac {n a -1}{a}}\right ) \left (y^{n}\right )^{-\frac {1}{a n}}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\textit {\_a}^{n}+\RootOf \left (-\textit {\_a}^{\frac {1}{a}} x^{-\frac {1}{a}} y^{n} \left (\textit {\_Z}^{n}\right )^{\frac {1}{a n}} \left (y^{n}\right )^{-\frac {1}{a n}}+\textit {\_a}^{\frac {1}{a}} x^{\frac {n a -1}{a}} \left (\textit {\_Z}^{n}\right )^{\frac {1}{a n}} \left (y^{n}\right )^{-\frac {1}{a n}}+\textit {\_Z}^{n}-\textit {\_a}^{n}\right )^{n}\right ) k n}{\left (a n \RootOf \left (-\textit {\_a}^{\frac {1}{a}} x^{-\frac {1}{a}} y^{n} \left (\textit {\_Z}^{n}\right )^{\frac {1}{a n}} \left (y^{n}\right )^{-\frac {1}{a n}}+\textit {\_a}^{\frac {1}{a}} x^{\frac {n a -1}{a}} \left (\textit {\_Z}^{n}\right )^{\frac {1}{a n}} \left (y^{n}\right )^{-\frac {1}{a n}}+\textit {\_Z}^{n}-\textit {\_a}^{n}\right )^{n}+\textit {\_a}^{n}-\RootOf \left (-\textit {\_a}^{\frac {1}{a}} x^{-\frac {1}{a}} y^{n} \left (\textit {\_Z}^{n}\right )^{\frac {1}{a n}} \left (y^{n}\right )^{-\frac {1}{a n}}+\textit {\_a}^{\frac {1}{a}} x^{\frac {n a -1}{a}} \left (\textit {\_Z}^{n}\right )^{\frac {1}{a n}} \left (y^{n}\right )^{-\frac {1}{a n}}+\textit {\_Z}^{n}-\textit {\_a}^{n}\right )^{n}\right ) \textit {\_a}}d \textit {\_a}}\]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x((n-2) y^n -2 x^n ) w_x + y(2 y^n - (n-2) x^n) w_y =\left ( (a(n-2)+2 b) y^n - (2 a + b(n-2)) x^n) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = x*((n - 2)*y^n - 2*x^n)*D[w[x, y], x] + y*(2*y^n - (n - 2)*x^n)*D[w[x, y], y] == ((a*(n - 2) + 2*b)*y^n - (2*a + b*(n - 2))*x^n)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := x*((n-2)*y^n -2*x^n )*diff(w(x,y),x)+y*(2*y^n - (n-2)*x^n)*diff(w(x,y),y) = ((a*(n-2)+2*b)*y^n - (2*a + b*(n-2))*x^n)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (x^{n}+y^{n}\right ) \left (y^{n}\right )^{-\frac {2}{n}}}{x^{2}}\right ) {\mathrm e}^{\int _{}^{x}\frac {2 \left (\frac {1}{2} b n +a -b \right ) \textit {\_a}^{n}+\left (-n a +2 a -2 b \right ) \RootOf \left (\textit {\_a}^{2} x^{n} \left (\textit {\_Z}^{n}\right )^{\frac {2}{n}} \left (y^{n}\right )^{-\frac {2}{n}}+\textit {\_a}^{2} y^{n} \left (\textit {\_Z}^{n}\right )^{\frac {2}{n}} \left (y^{n}\right )^{-\frac {2}{n}}-x^{2} \textit {\_Z}^{n}-x^{2} \textit {\_a}^{n}\right )^{n}}{\left (-n \RootOf \left (\textit {\_a}^{2} x^{n} \left (\textit {\_Z}^{n}\right )^{\frac {2}{n}} \left (y^{n}\right )^{-\frac {2}{n}}+\textit {\_a}^{2} y^{n} \left (\textit {\_Z}^{n}\right )^{\frac {2}{n}} \left (y^{n}\right )^{-\frac {2}{n}}-x^{2} \textit {\_Z}^{n}-x^{2} \textit {\_a}^{n}\right )^{n}+2 \textit {\_a}^{n}+2 \RootOf \left (\textit {\_a}^{2} x^{n} \left (\textit {\_Z}^{n}\right )^{\frac {2}{n}} \left (y^{n}\right )^{-\frac {2}{n}}+\textit {\_a}^{2} y^{n} \left (\textit {\_Z}^{n}\right )^{\frac {2}{n}} \left (y^{n}\right )^{-\frac {2}{n}}-x^{2} \textit {\_Z}^{n}-x^{2} \textit {\_a}^{n}\right )^{n}\right ) \textit {\_a}}d \textit {\_a}}\]
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