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]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )-\frac {a \beta (2 b+c y)+c (b \beta x+\beta c x y+c \gamma )}{c^3}\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =1/8\,{\frac {1}{{c}^{3}} \left ( 8\,{{\rm e}^{{\frac {cx}{a}}}}{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {c}^{3}+ \left ( -8\,\beta \,xy-1 \right ) {c}^{2}-8\,\beta \, \left ( ya+bx \right ) c-16\,ab\beta \right ) }\]
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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'));
\[w \left ( x,y \right ) =1/8\,{\frac {1}{{c}^{3}} \left ( 8\,{{\rm e}^{{\frac {cx}{a}}}}{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {c}^{3}+ \left ( -8\,\beta \,{x}^{2}-xy-8\,\delta \right ) {c}^{2}+ \left ( \left ( -16\,\beta \,x-y \right ) a-bx \right ) c-16\,\beta \,{a}^{2}-2\,ab \right ) }\]
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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'));
\[w \left ( x,y \right ) ={\frac {y}{8\,b-8\,c+8\,a}{x}^{{\frac {a+b}{a}}-{\frac {b}{a}}}}+{\frac {\beta \,{x}^{2}}{2\,a-c}}-{\frac {\delta }{c}}+{x}^{{\frac {c}{a}}}{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
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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'));
\[\text {Expression too large to display}\]
____________________________________________________________________________________
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]];
\begin {align*} & \left \{w(x,y)\to e^{\frac {a \text {c1} y^3+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}}{\sqrt [3]{a} \text {b1} \left (a y^3\right )^{2/3}}} \left (c_1\left (-\frac {\text {b1} x^3+3 \text {b0} x-a y^3}{3 a}\right )+\int _1^x\frac {e^{-\frac {\text {c1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[1]^3+3 \text {b0} K[1]\right )+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}}{\sqrt [3]{a} \text {b1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[1]^3+3 \text {b0} K[1]\right )^{2/3}}} \left (\text {s1} K[1]^2+\text {s0}\right )}{\sqrt [3]{a} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[1]^3+3 \text {b0} K[1]\right )^{2/3}}dK[1]\right )\right \}\\& \left \{w(x,y)\to e^{\frac {\sqrt [3]{-1} \left (-a \text {c1} y^3-3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (a y^3\right )^{2/3}}} \left (c_1\left (-\frac {\text {b1} x^3+3 \text {b0} x-a y^3}{3 a}\right )+\int _1^x-\frac {\sqrt [3]{-1} e^{\frac {\sqrt [3]{-1} \left (\text {c1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[2]^3+3 \text {b0} K[2]\right )+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[2]^3+3 \text {b0} K[2]\right )^{2/3}}} \left (\text {s1} K[2]^2+\text {s0}\right )}{\sqrt [3]{a} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[2]^3+3 \text {b0} K[2]\right )^{2/3}}dK[2]\right )\right \}\\& \left \{w(x,y)\to e^{\frac {(-1)^{2/3} \left (a \text {c1} y^3+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (a y^3\right )^{2/3}}} \left (c_1\left (-\frac {\text {b1} x^3+3 \text {b0} x-a y^3}{3 a}\right )+\int _1^x\frac {(-1)^{2/3} e^{-\frac {(-1)^{2/3} \left (\text {c1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[3]^3+3 \text {b0} K[3]\right )+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[3]^3+3 \text {b0} K[3]\right )^{2/3}}} \left (\text {s1} K[3]^2+\text {s0}\right )}{\sqrt [3]{a} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[3]^3+3 \text {b0} K[3]\right )^{2/3}}dK[3]\right )\right \}\\ \end {align*}
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}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}-\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\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}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f} \right ) ^{2}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}y-{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\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 ) }}}+{\it k12}\,{\it \_f}\, \left ( y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}-\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\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}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f} \right ) {{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}y-{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{{\rm e}^{-\int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}y-{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\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}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{{{\it \_b}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}y-{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0}}{{{\it \_b}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\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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