Added March 10, 2019.
Problem Chapter 5.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \beta x y+\gamma \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + beta*x*y + gamma; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {a \beta (2 b+c y)+c (b \beta x+\beta c x y+c \gamma )}{c^3}+e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+beta*x*y+gamma; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{{c}^{3}} \left ( {{\rm e}^{{\frac {cx}{a}}}}{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {c}^{3}+ \left ( -\beta \,xy-\gamma \right ) {c}^{2}-\beta \, \left ( ay+xb \right ) c-2\,a\beta \,b \right ) }\]
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Added March 10, 2019.
Problem Chapter 5.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + x(\beta x+\gamma y)+\delta \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + x*(beta*x + gamma*y) + delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {c^3 \left (-e^{\frac {c x}{a}}\right ) c_1\left (y-\frac {b x}{a}\right )+2 a^2 \beta +a (2 b \gamma +2 \beta c x+c \gamma y)+c \left (b \gamma x+c \left (\beta x^2+\delta +\gamma x y\right )\right )}{c^3}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+x*(beta*x+gamma*y)+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{{c}^{3}} \left ( {{\rm e}^{{\frac {cx}{a}}}}{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {c}^{3}+ \left ( -\beta \,{x}^{2}-\gamma \,xy-\delta \right ) {c}^{2}+ \left ( \left ( -ay-xb \right ) \gamma -2\,a\beta \,x \right ) c-2\,{a}^{2}\beta -2\,a\gamma \,b \right ) }\]
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Added March 10, 2019.
Problem Chapter 5.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = w + a x^2+b y^2+c \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == w[x, y] + a*x^2 + b*y^2 + c; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to x c_1\left (\frac {y}{x}\right )+a x^2+b y^2-c\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = w(x,y)+a*x^2+b*y^2+c; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) =b{y}^{2}+a{x}^{2}+{\it \_F1} \left ( {\frac {y}{x}} \right ) x-c\]
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Added March 10, 2019.
Problem Chapter 5.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c w + x(\beta x+\gamma y)+ \delta \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x, y] + x*(beta*x + gamma*y) + delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {c (2 a-c) (a+b-c) x^{\frac {c}{a}} c_1\left (y x^{-\frac {b}{a}}\right )-2 a^2 \delta -2 a b \delta +a c (x (\beta x+2 \gamma y)+3 \delta )+b c \left (\beta x^2+\delta \right )-c^2 (x (\beta x+\gamma y)+\delta )}{c (c-2 a) (-a-b+c)}\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*w(x,y)+x*(beta*x+gamma*y)+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {\gamma \,y}{-c+b+a}{x}^{{\frac {a+b}{a}}-{\frac {b}{a}}}}+{\frac {\beta \,{x}^{2}}{2\,a-c}}-{\frac {\delta }{c}}+{x}^{{\frac {c}{a}}}{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
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Added March 10, 2019.
Problem Chapter 5.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + (b_2 x^2+b_1 x+b_0) w_y = (c_2 x^2+c_1 x+c_0) w + s_2 x^2+s_1 x+s_0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*y*D[w[x, y], x] + (b2*x^2 + b1*x + b0)*D[w[x, y], y] == (c2*x^2 + c1*x + c0)*w[x, y] + s2*x^2 + s1*x + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✓
restart; pde := a*y*diff(w(x,y),x)+ (b2*x^2+b1*x+b0)*diff(w(x,y),y) = (c2*x^2+c1*x+c0)*w(x,y)+s2*x^2+s1*x+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[\text {Expression too large to display}\]
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Added March 10, 2019.
Problem Chapter 5.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^2 w_x + (b_1 x^2+b_0) w_y = (c_1 x^2+c_0) w + s_1 x^2+s_0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*y^2*D[w[x, y], x] + (b1*x^2 + b0)*D[w[x, y], y] == (c1*x^2 + c0)*w[x, y] + s1*x^2 + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✓
restart; pde := a*y*diff(w(x,y),x)+ (b1*x^2+b0)*diff(w(x,y),y) = (c1*x^2+c0)*w(x,y)+s1*x^2+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[\text {Expression too large to display}\]
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Added March 10, 2019.
Problem Chapter 5.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x^2+a_0) w_x + (y+b_2 x^2+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_{22}y^2+k{12} x y+k_{11} x^2+k_0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a1*x^2 + a0)*y^2*D[w[x, y], x] + (y + b2*x^2 + b1*x + b0)*D[w[x, y], y] == (c2*y + c1*x + c0)*w[x, y] + k22*y^2 + k12*x*y + k11*x^2 + k0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a1*x^2+a0)*diff(w(x,y),x)+ (y+b2*x^2+b1*x+b0)*diff(w(x,y),y) = (c2*y+c1*x+c0)*w(x,y)+ k22*y^2+k12*x*y+k11*x^2+k0; 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}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it k22}\, \left ( y{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}-\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}{\it \_f} \right ) ^{2}{{\rm e}^{-{ \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}y-{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}{\it \_f}+{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-2\,\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}+{\it k12}\,{\it \_f}\, \left ( y{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}-\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}{\it \_f} \right ) {{\rm e}^{-{ \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}y-{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}{\it \_f}+{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}+{{\rm e}^{-\int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}y-{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,{\it \_f}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}{\it \_f}+{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}}} \left ( {\it k11}\,{{\it \_f}}^{2}+{\it k0} \right ) \right ) }{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+y{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{{{\it \_b}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_b}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}y-{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_b}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,x{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\arctan \left ( {{\it a1}\,{\it \_b}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}\int \!{\frac {{{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0}}{{{\it \_b}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\arctan \left ( {{\it a1}\,{\it \_b}{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}} \right ) {\frac {1}{\sqrt {{\it a0}\,{\it a1}}}}}}}}\,{\rm d}{\it \_b}+{\it c1}\,{\it \_b}+{\it c0} \right ) }{d{\it \_b}}}}\]
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Added March 10, 2019.
Problem Chapter 5.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x^2+a_0) w_x + (b_2 y^2+b_1 x y) w_y = (c_2 y^2+c_1 x^2) w + s_{22}y^2+s_{12} x y+s_{11} x^2+s_0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a1*x^2 + a0)*y^2*D[w[x, y], x] + (b2*y^2 + b1*x^2)*D[w[x, y], y] == (c2*y^2 + c1*x^2)*w[x, y] + s22*y^2 + s12*x*y + s11*x^2 + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a1*x^2+a0)*diff(w(x,y),x)+ (b2*y^2+b1*x^2)*diff(w(x,y),y) = (c2*y^2+c1*x^2)*w(x,y)+ s22*y^2+s12*x*y+s11*x^2+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[\text {Expression too large to display}\]
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