6.5.2 2.2

6.5.2.1 [1204] Problem 1

problem number 1204

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]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} (b \beta K[1] (K[1]-x)+a (\gamma +\beta y K[1]))}{a^2}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  c*w(x,y)+beta*x*y+gamma; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {y a -x b}{a}\right ) {\mathrm e}^{\frac {c x}{a}} c^{3}+\left (-\beta x y -\gamma \right ) c^{2}-\beta \left (y a +x b \right ) c -2 a b \beta }{c^{3}}\]

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6.5.2.2 [1205] Problem 2

problem number 1205

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]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \left (b \gamma K[1] (K[1]-x)+a \left (\beta K[1]^2+\gamma y K[1]+\delta \right )\right )}{a^2}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  c*w(x,y)+x*(beta*x+gamma*y)+delta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {y a -b x}{a}\right ) {\mathrm e}^{\frac {c x}{a}} c^{3}+\left (-\beta \,x^{2}-\gamma x y -\delta \right ) c^{2}+\left (\left (-y a -b x \right ) \gamma -2 a \beta x \right ) c -2 a^{2} \beta -2 a b \gamma }{c^{3}}\]

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6.5.2.3 [1206] Problem 3

problem number 1206

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'));
 

\[w \left (x , y\right ) = b \,y^{2}+a \,x^{2}+f_{1} \left (\frac {y}{x}\right ) x -c\]

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6.5.2.4 [1207] Problem 4

problem number 1207

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'));
 

\[w \left (x , y\right ) = \frac {\gamma y x}{a +b -c}+\frac {x^{2} \beta }{2 a -c}-\frac {\delta }{c}+x^{\frac {c}{a}} f_{1} \left (y \,x^{-\frac {b}{a}}\right )\]

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6.5.2.5 [1208] Problem 5

problem number 1208

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]];
 

\begin{align*}& \left \{w(x,y)\to e^{-\frac {a \text {c2} y^2+2 \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-x}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right ),\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {b2} \text {c0}-\text {b0} \text {c2}+(\text {b2} \text {c1}-\text {b1} \text {c2}) \text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]\right )-(\text {b2} \text {c1}-\text {b1} \text {c2}) E\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-x}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right )|\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )\right ) \sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-x}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}} \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right ) \sqrt {\frac {\left (x-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-x\right )}{\left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )^2}}}{\sqrt {a} \text {b2} \sqrt {a y^2}}} \left (c_1\left (-\frac {2 \text {b2} x^3+3 \text {b1} x^2+6 \text {b0} x-3 a y^2}{6 a}\right )+\int _1^x-\frac {\sqrt {3} e^{\frac {\text {c2} \left (-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} K[1]^3+3 a y^2+3 \text {b1} K[1]^2+6 \text {b0} K[1]\right )+6 \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-K[1]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right ),\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {b2} \text {c0}-\text {b0} \text {c2}+(\text {b2} \text {c1}-\text {b1} \text {c2}) \text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]\right )-(\text {b2} \text {c1}-\text {b1} \text {c2}) E\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-K[1]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right )|\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )\right ) \sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-K[1]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}} \sqrt {-\frac {\left (K[1]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[1]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )}{\left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )^2}} \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )}{\sqrt {3} \sqrt {a} \text {b2} \sqrt {-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} K[1]^3+3 a y^2+3 \text {b1} K[1]^2+6 \text {b0} K[1]}}} (\text {s0}+K[1] (\text {s1}+\text {s2} K[1]))}{\sqrt {a} \sqrt {-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} K[1]^3+3 a y^2+3 \text {b1} K[1]^2+6 \text {b0} K[1]}}dK[1]\right )\right \}\\& \left \{w(x,y)\to e^{\frac {a \text {c2} y^2+2 \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-x}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right ),\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {b2} \text {c0}-\text {b0} \text {c2}+(\text {b2} \text {c1}-\text {b1} \text {c2}) \text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]\right )-(\text {b2} \text {c1}-\text {b1} \text {c2}) E\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-x}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right )|\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )\right ) \sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-x}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}} \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right ) \sqrt {\frac {\left (x-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-x\right )}{\left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )^2}}}{\sqrt {a} \text {b2} \sqrt {a y^2}}} \left (c_1\left (-\frac {2 \text {b2} x^3+3 \text {b1} x^2+6 \text {b0} x-3 a y^2}{6 a}\right )+\int _1^x\frac {\sqrt {3} e^{-\frac {\text {c2} \left (-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} K[2]^3+3 a y^2+3 \text {b1} K[2]^2+6 \text {b0} K[2]\right )+6 \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-K[2]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right ),\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {b2} \text {c0}-\text {b0} \text {c2}+(\text {b2} \text {c1}-\text {b1} \text {c2}) \text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]\right )-(\text {b2} \text {c1}-\text {b1} \text {c2}) E\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-K[2]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]}}\right )|\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}\right ) \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )\right ) \sqrt {\frac {\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-K[2]}{\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]}} \sqrt {-\frac {\left (K[2]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[2]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )}{\left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )^2}} \left (\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,2\right ]-\text {Root}\left [-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} \text {$\#$1}^3+3 a y^2+3 \text {b1} \text {$\#$1}^2+6 \text {b0} \text {$\#$1}\&,3\right ]\right )}{\sqrt {3} \sqrt {a} \text {b2} \sqrt {-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} K[2]^3+3 a y^2+3 \text {b1} K[2]^2+6 \text {b0} K[2]}}} (\text {s0}+K[2] (\text {s1}+\text {s2} K[2]))}{\sqrt {a} \sqrt {-2 \text {b2} x^3-3 \text {b1} x^2-6 \text {b0} x+2 \text {b2} K[2]^3+3 a y^2+3 \text {b1} K[2]^2+6 \text {b0} K[2]}}dK[2]\right )\right \}\\\end{align*}

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'));
 

time expired

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6.5.2.6 [1209] Problem 6

problem number 1209

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}) \operatorname {Hypergeometric2F1}\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 {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}}{\sqrt [3]{a} \text {b1} \left (a y^3\right )^{2/3}}} \left (c_1\left (-\frac {\text {b1} x^3+3 \text {b0} x-a y^3}{3 a}\right )+\int _1^x\frac {e^{-\frac {\text {c1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[1]^3+3 \text {b0} K[1]\right )+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\left (K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {K[1]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}}{\sqrt [3]{a} \text {b1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[1]^3+3 \text {b0} K[1]\right )^{2/3}}} \left (\text {s1} K[1]^2+\text {s0}\right )}{\sqrt [3]{a} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[1]^3+3 \text {b0} K[1]\right )^{2/3}}dK[1]\right )\right \}\\& \left \{w(x,y)\to e^{\frac {\sqrt [3]{-1} \left (-a \text {c1} y^3-3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (a y^3\right )^{2/3}}} \left (c_1\left (-\frac {\text {b1} x^3+3 \text {b0} x-a y^3}{3 a}\right )+\int _1^x-\frac {\sqrt [3]{-1} e^{\frac {\sqrt [3]{-1} \left (\text {c1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[2]^3+3 \text {b0} K[2]\right )+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\left (K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {K[2]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[2]^3+3 \text {b0} K[2]\right )^{2/3}}} \left (\text {s1} K[2]^2+\text {s0}\right )}{\sqrt [3]{a} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[2]^3+3 \text {b0} K[2]\right )^{2/3}}dK[2]\right )\right \}\\& \left \{w(x,y)\to e^{\frac {(-1)^{2/3} \left (a \text {c1} y^3+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {x-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}}\right )}{\sqrt [3]{a} \text {b1} \left (a y^3\right )^{2/3}}} \left (c_1\left (-\frac {\text {b1} x^3+3 \text {b0} x-a y^3}{3 a}\right )+\int _1^x\frac {(-1)^{2/3} e^{-\frac {(-1)^{2/3} \left (\text {c1} \left (-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} K[3]^3+3 \text {b0} K[3]\right )+3 (\text {b1} \text {c0}-\text {b0} \text {c1}) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\left (K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right )}{\left (\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]\right ) \left (K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]\right )}\right ) \left (K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]\right ) \left (\frac {K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,2\right ]}\right )^{2/3} \sqrt [3]{\frac {K[3]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,3\right ]}{\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$1}\&,1\right ]-\text {Root}\left [-\text {b1} x^3-3 \text {b0} x+a y^3+\text {b1} \text {$\#$1}^3+3 \text {b0} \text {$\#$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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6.5.2.7 [1210] Problem 7

problem number 1210

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'));
 

\[\text {Expression too large to display}\]

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6.5.2.8 [1211] Problem 8

problem number 1211

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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