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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {ad \left ( n+1 \right ) {y}^{m+1}+{x}^{n+1}bc \left ( m+1 \right ) }{a \left ( n+1 \right ) b \left ( m+1 \right ) }}}}\]
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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;
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {c \left ( ay \left ( n+2 \right ) {x}^{n+1}-{x}^{n+2}b \right ) }{ \left ( n+2 \right ) \left ( n+1 \right ) {a}^{2}}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {x}^{{\frac { \left ( ya-bx \right ) c}{{a}^{2}}}}{{\rm e}^{{\frac {cxb}{{a}^{2}}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{-{\frac { \left ( ya-bx \right ) c}{{a}^{2}x}}}}{x}^{{\frac {bc}{{a}^{2}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) {{\rm e}^{1/2\,{\frac {a \left ( {x}^{2}+{y}^{2} \right ) ^{k}}{k}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {{\rm e}^{{\frac {c{x}^{n}{y}^{m}}{an+bm}}}}\]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {{\rm e}^{{\frac {c{x}^{n}bm+k{y}^{m}an}{abmn}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {n}{m}}} \right ) {{\rm e}^{{\frac { \left ( {x}^{n}a+{y}^{m}b \right ) ^{k}}{knm}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{-n+1}b \left ( m-1 \right ) +a \left ( n-1 \right ) {y}^{-m+1}}{a \left ( n-1 \right ) }} \right ) {{\rm e}^{{\frac {1}{ab \left ( k-n+1 \right ) \left ( m-s-1 \right ) } \left ( -{a}^{2}{y}^{-m+1}{a}^{{\frac {s}{m-1}}-1} \left ( n-1 \right ) ^{{\frac {s}{m-1}}}{{\rm e}^{{\frac {-i\pi \,s}{2\,m-2} \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) -{\it csgn} \left ( {\frac {i}{n-1}} \right ) \right ) {\it csgn} \left ( i \left ( n-1 \right ) {y}^{-m+1}a \right ) - \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) \right ) ^{3}+{\it csgn} \left ( {\frac {i}{n-1}} \right ) \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) \right ) ^{2}+{\it csgn} \left ( i{y}^{-m+1} \right ) \left ( {\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ( {\frac {i}{a}} \right ) \right ) {\it csgn} \left ( i{y}^{-m+1}a \right ) - \left ( {\it csgn} \left ( i{y}^{-m+1} \right ) \right ) ^{2} \left ( {\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ( {\frac {i}{a}} \right ) \right ) \right ) }}}d \left ( k-n+1 \right ) \left ( a \left ( n-1 \right ) {y}^{-m+1} \right ) ^{-{\frac {s}{m-1}}}+{x}^{k-n+1}cb \left ( m-s-1 \right ) \right ) }}}\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\frac {x^{1-n} \left (\frac {d}{1-n}-\frac {c x^k y^s e^{-\frac {b s x^{m-n+1}}{a m-a n+a}} \left (-\frac {b s x^{m-n+1}}{a m-a n+a}\right )^{\frac {-k+n-1}{m-n+1}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a m-a n+a}\right )}{m-n+1}\right )}{a}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-n}}{a} \left ( c{{\it \_a}}^{k} \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{-n+m+1}-{{\it \_a}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{s}+d \right ) }{d{\it \_a}}}}\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (c (a (m-n+1))^{\frac {k-m}{m-n+1}} b^{\frac {-k+n-1}{m-n+1}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\int _1^x\frac {K[1]^{-n} \left (s \left (b^{-\frac {k+1}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}-b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}+b^{\frac {k+1}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^p+d\right )}{a}dK[1]\right )\right \}\right \}\]
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 ) ={\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 ( {x}^{k-n+1}b+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( -n+m+1 \right ) ^{2} \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+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{-n+p} \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 ( b{{\it \_a}}^{k-n+1}+a{{\it \_a}}^{k-m} \left ( k+m-2\,n+2 \right ) \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{{\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 ( {x}^{k-n+1}b+a{x}^{k-m} \left ( k+m-2\,n+2 \right ) \right ) \left ( -n+m+1 \right ) ^{2} \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+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) {{\rm e}^{{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) } \right ) ^{q}s+{{\it \_a}}^{-n}d \right ) }{d{\it \_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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{x}^{-n+m+1}b \left ( k-1 \right ) +{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-n}}{a} \left ( c{{\it \_a}}^{p} \left ( \left ( {\frac {{x}^{-n+m+1}b \left ( k-1 \right ) -b \left ( k-1 \right ) {{\it \_a}}^{-n+m+1}+{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) ^{q}+s \right ) }{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) {{\rm e}^{\int ^{x}\!{\frac {c{{\it \_a}}^{m}+s}{a} \left ( \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) +b{{\it \_a}}^{n+1} \left ( k+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( - \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}} \left ( {y}^{n}{x}^{-{a}^{-1}}-{x}^{{\frac {an-1}{a}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {kn}{{\it \_a}} \left ( {{\it \_a}}^{n}+ \left ( \RootOf \left ( -{y}^{n} \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}{x}^{-{a}^{-1}}+ \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}{x}^{{\frac {an-1}{a}}}-{{\it \_a}}^{n}+{{\it \_Z}}^{n} \right ) \right ) ^{n} \right ) \left ( \left ( \RootOf \left ( -{y}^{n} \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}{x}^{-{a}^{-1}}+ \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}{x}^{{\frac {an-1}{a}}}-{{\it \_a}}^{n}+{{\it \_Z}}^{n} \right ) \right ) ^{n}an+{{\it \_a}}^{n}- \left ( \RootOf \left ( -{y}^{n} \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}{x}^{-{a}^{-1}}+ \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}{x}^{{\frac {an-1}{a}}}-{{\it \_a}}^{n}+{{\it \_Z}}^{n} \right ) \right ) ^{n} \right ) ^{-1}}{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{x}^{n}+{y}^{n}}{{x}^{2}} \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}}} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{{\it \_a}} \left ( \left ( -an+2\,a-2\,b \right ) \left ( \RootOf \left ( \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}} \left ( {{\it \_Z}}^{n} \right ) ^{2\,{n}^{-1}}{{\it \_a}}^{2}{x}^{n}+ \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}} \left ( {{\it \_Z}}^{n} \right ) ^{2\,{n}^{-1}}{{\it \_a}}^{2}{y}^{n}-{{\it \_a}}^{n}{x}^{2}-{{\it \_Z}}^{n}{x}^{2} \right ) \right ) ^{n}+2\, \left ( 1/2\,bn+a-b \right ) {{\it \_a}}^{n} \right ) \left ( - \left ( \RootOf \left ( \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}} \left ( {{\it \_Z}}^{n} \right ) ^{2\,{n}^{-1}}{{\it \_a}}^{2}{x}^{n}+ \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}} \left ( {{\it \_Z}}^{n} \right ) ^{2\,{n}^{-1}}{{\it \_a}}^{2}{y}^{n}-{{\it \_a}}^{n}{x}^{2}-{{\it \_Z}}^{n}{x}^{2} \right ) \right ) ^{n}n+2\,{{\it \_a}}^{n}+2\, \left ( \RootOf \left ( \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}} \left ( {{\it \_Z}}^{n} \right ) ^{2\,{n}^{-1}}{{\it \_a}}^{2}{x}^{n}+ \left ( {y}^{n} \right ) ^{-2\,{n}^{-1}} \left ( {{\it \_Z}}^{n} \right ) ^{2\,{n}^{-1}}{{\it \_a}}^{2}{y}^{n}-{{\it \_a}}^{n}{x}^{2}-{{\it \_Z}}^{n}{x}^{2} \right ) \right ) ^{n} \right ) ^{-1}}{d{\it \_a}}}}\]
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