7.5.2 2.2

7.5.2.1 [1206] Problem 1

problem number 1206

Added March 10, 2019.

Problem Chapter 5.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=cw+\beta xy+\gamma$$

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]];
 

$$\left\{\{w(x,y)\to -\frac{a\beta (2b+cy)+c(b\beta x+\beta cxy+c\gamma )}{c^3}+e^{\frac{cx}{a}}{c}_1(y-\frac{bx}{a})\}\right\}$$

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'));
 

$$w(x,y)=\frac{c^3\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right){\mathrm{e}}^{\frac{cx}{a}}-2ab\beta -(ay+bx)\beta c+(-\beta xy-\gamma )c^2}{c^3}$$

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7.5.2.2 [1207] Problem 2

problem number 1207

Added March 10, 2019.

Problem Chapter 5.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=cw+x(\beta x+\gamma y)+\delta$$

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]];
 

$$\left\{\{w(x,y)\to -\frac{c^3(-e^{\frac{cx}{a}})c_1(y-\frac{bx}{a})+2a^2\beta +a(2b\gamma +2\beta cx+c\gamma y)+c(b\gamma x+c(\beta x^2+\delta +\gamma xy))}{c^3}\}\right\}$$

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'));
 

$$w(x,y)=\frac{c^3\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right){\mathrm{e}}^{\frac{cx}{a}}-2a^2\beta -2\gamma ab+(-\beta x^2-\gamma xy-\delta )c^2+(-2a\beta x+(-ay-bx)\gamma )c}{c^3}$$

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7.5.2.3 [1208] Problem 3

problem number 1208

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]];
 

$$\left\{\{w(x,y)\to x{c}_1\left(\frac{y}{x}\right)+a{x}^2+b{y}^2-c\}\right\}$$

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'));
 

$$w(x,y)=a{x}^2+b{y}^2+x\mathit{\text{\_F1}}\left(\frac{y}{x}\right)-c$$

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7.5.2.4 [1209] Problem 4

problem number 1209

Added March 10, 2019.

Problem Chapter 5.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$ax{w}_x+by{w}_y=cw+x(\beta x+\gamma y)+\delta$$

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]];
 

$$\left\{\{w(x,y)\to \frac{c(2a-c)(a+b-c)x^{\frac{c}{a}}{c}_1\left(y{x}^{-\frac{b}{a}}\right)-2a^2\delta -2ab\delta +ac(x(\beta x+2\gamma y)+3\delta )+bc(\beta x^2+\delta )-c^2(x(\beta x+\gamma y)+\delta )}{c(c-2a)(-a-b+c)}\}\right\}$$

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'));
 

$$w(x,y)=\frac{\beta x^2}{2a-c}+\frac{\gamma y{x}^{-\frac{b}{a}+\frac{a+b}{a}}}{a+b-c}+x^{\frac{c}{a}}\mathit{\text{\_F1}}\left(y{x}^{-\frac{b}{a}}\right)-\frac{\delta }{c}$$

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7.5.2.5 [1210] Problem 5

problem number 1210

Added March 10, 2019.

Problem Chapter 5.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$ay{w}_x+(b_2{x}^2+b_{1}x+b_0)w_y=(c_2{x}^2+c_{1}x+c_0)w+s_2{x}^2+s_{1}x+s_0$$

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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7.5.2.6 [1211] Problem 6

problem number 1211

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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7.5.2.7 [1212] Problem 7

problem number 1212

Added March 10, 2019.

Problem Chapter 5.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$(a_1{x}^2+a_0)w_x+(y+b_2{x}^2+b_{1}x+b_0)w_y=(c_{2}y+c_{1}x+c_0)w+k_{22}{y}^2+k12xy+k_{11}{x}^2+k_0$$

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(x,y)=({\int }_{}^x\frac{(y{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\int \frac{({\mathit{\text{\_f}}}^2\mathit{b}\mathit{2}+\mathit{\text{\_f}}\mathit{b}\mathit{1}+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}}-(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx))\mathit{\text{\_f}}\mathit{k}\mathit{12}{\mathrm{e}}^{-\frac{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}(\int \frac{\mathit{c}\mathit{2}y{\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{c}\mathit{2}(\int \frac{({\mathit{\text{\_f}}}^2\mathit{b}\mathit{2}+\mathit{\text{\_f}}\mathit{b}\mathit{1}+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}}){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}-\mathit{c}\mathit{2}(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{\text{\_f}}\mathit{c}\mathit{1}+\mathit{c}\mathit{0}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}})-\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+{(y{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\int \frac{({\mathit{\text{\_f}}}^2\mathit{b}\mathit{2}+\mathit{\text{\_f}}\mathit{b}\mathit{1}+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}}-(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx))}^2\mathit{k}\mathit{22}{\mathrm{e}}^{-\frac{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}(\int \frac{\mathit{c}\mathit{2}y{\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{c}\mathit{2}(\int \frac{({\mathit{\text{\_f}}}^2\mathit{b}\mathit{2}+\mathit{\text{\_f}}\mathit{b}\mathit{1}+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}}){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}-\mathit{c}\mathit{2}(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{\text{\_f}}\mathit{c}\mathit{1}+\mathit{c}\mathit{0}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}})-2\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+(\mathit{k}\mathit{11}{\mathit{\text{\_f}}}^2+\mathit{k}\mathit{0}){\mathrm{e}}^{-(\int \frac{\mathit{c}\mathit{2}y{\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{c}\mathit{2}(\int \frac{({\mathit{\text{\_f}}}^2\mathit{b}\mathit{2}+\mathit{\text{\_f}}\mathit{b}\mathit{1}+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}}){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}-\mathit{c}\mathit{2}(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_f}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{\text{\_f}}\mathit{c}\mathit{1}+\mathit{c}\mathit{0}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}})}}{{\mathit{\text{\_f}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_f}}+\mathit{\text{\_F1}}(y{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}-(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx))){\mathrm{e}}^{{\int }_{}^x\frac{\mathit{c}\mathit{2}y{\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_b}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}{\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{c}\mathit{2}(\int \frac{({\mathit{\text{\_b}}}^2\mathit{b}\mathit{2}+\mathit{\text{\_b}}\mathit{b}\mathit{1}+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{\text{\_b}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{{\mathit{\text{\_b}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_b}}){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_b}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}-\mathit{c}\mathit{2}(\int \frac{(\mathit{b}\mathit{2}{x}^2+\mathit{b}\mathit{1}x+\mathit{b}\mathit{0}){\mathrm{e}}^{-\frac{\arctan \left(\frac{\mathit{a}\mathit{1}x}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}}{\mathit{a}\mathit{1}{x}^2+\mathit{a}\mathit{0}}dx){\mathrm{e}}^{\frac{\arctan \left(\frac{\mathit{\text{\_b}}\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}\right)}{\sqrt{\mathit{a}\mathit{0}\mathit{a}\mathit{1}}}}+\mathit{\text{\_b}}\mathit{c}\mathit{1}+\mathit{c}\mathit{0}}{{\mathit{\text{\_b}}}^2\mathit{a}\mathit{1}+\mathit{a}\mathit{0}}d\mathit{\text{\_b}}}$$

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7.5.2.8 [1213] Problem 8

problem number 1213

Added March 10, 2019.

Problem Chapter 5.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$(a_1{x}^2+a_0)w_x+(b_2{y}^2+b_{1}xy)w_y=(c_2{y}^2+c_1{x}^2)w+s_{22}{y}^2+s_{12}xy+s_{11}{x}^2+s_0$$

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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