7.5.2 2.2

7.5.2.1 [1206] Problem 1
7.5.2.2 [1207] Problem 2
7.5.2.3 [1208] Problem 3
7.5.2.4 [1209] Problem 4
7.5.2.5 [1210] Problem 5
7.5.2.6 [1211] Problem 6
7.5.2.7 [1212] Problem 7
7.5.2.8 [1213] Problem 8

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 \(w(x,y)\)

\[ a w_x + b w_y = c w + \beta x y+\gamma \]

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'));
 

\[w \left (x , y\right ) = \frac {c^{3} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {c x}{a}}-2 a b \beta -\left (a y +b x \right ) \beta c +\left (-\beta x y -\gamma \right ) c^{2}}{c^{3}}\]

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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 \(w(x,y)\)

\[ a w_x + b w_y = c w + x(\beta x+\gamma y)+\delta \]

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'));
 

\[w \left (x , y\right ) = \frac {c^{3} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {c x}{a}}-2 a^{2} \beta -2 \gamma a b +\left (-\beta \,x^{2}-\gamma x y -\delta \right ) c^{2}+\left (-2 a \beta x +\left (-a y -b x \right ) \gamma \right ) c}{c^{3}}\]

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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]];
 

\[\left \{\left \{w(x,y)\to x c_1\left (\frac {y}{x}\right )+a x^2+b y^2-c\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = a \,x^{2}+b \,y^{2}+x \textit {\_F1} \left (\frac {y}{x}\right )-c\]

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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 \(w(x,y)\)

\[ a x w_x + b y w_y = c w + x(\beta x+\gamma y)+ \delta \]

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'));
 

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

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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 \(w(x,y)\)

\[ a y w_x + (b_2 x^2+b_1 x+b_0) w_y = (c_2 x^2+c_1 x+c_0) w + s_2 x^2+s_1 x+s_0 \]

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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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 \(w(x,y)\)

\[ (a_1 x^2+a_0) w_x + (y+b_2 x^2+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_{22}y^2+k{12} x y+k_{11} x^2+k_0 \]

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 {\left (y \,{\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\int \frac {\left (\textit {\_f}^{2} \mathit {b2} +\textit {\_f} \mathit {b1} +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} -\left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right )\right ) \textit {\_f} \mathit {k12} \,{\mathrm e}^{-\frac {\sqrt {\mathit {a0} \mathit {a1}}\, \left (\int \frac {\mathit {c2} y \,{\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}} {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\mathit {c2} \left (\int \frac {\left (\textit {\_f}^{2} \mathit {b2} +\textit {\_f} \mathit {b1} +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}-\mathit {c2} \left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\textit {\_f} \mathit {c1} +\mathit {c0}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} \right )-\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\left (y \,{\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\int \frac {\left (\textit {\_f}^{2} \mathit {b2} +\textit {\_f} \mathit {b1} +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} -\left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right )\right )^{2} \mathit {k22} \,{\mathrm e}^{-\frac {\sqrt {\mathit {a0} \mathit {a1}}\, \left (\int \frac {\mathit {c2} y \,{\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}} {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\mathit {c2} \left (\int \frac {\left (\textit {\_f}^{2} \mathit {b2} +\textit {\_f} \mathit {b1} +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}-\mathit {c2} \left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\textit {\_f} \mathit {c1} +\mathit {c0}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} \right )-2 \arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\left (\mathit {k11} \,\textit {\_f}^{2}+\mathit {k0} \right ) {\mathrm e}^{-\left (\int \frac {\mathit {c2} y \,{\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}} {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\mathit {c2} \left (\int \frac {\left (\textit {\_f}^{2} \mathit {b2} +\textit {\_f} \mathit {b1} +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}-\mathit {c2} \left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_f} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\textit {\_f} \mathit {c1} +\mathit {c0}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} \right )}}{\textit {\_f}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_f} +\textit {\_F1} \left (y \,{\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}-\left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {c2} y \,{\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_b} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}} {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\mathit {c2} \left (\int \frac {\left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\textit {\_b} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\textit {\_b}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_b} \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_b} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}-\mathit {c2} \left (\int \frac {\left (\mathit {b2} \,x^{2}+\mathit {b1} x +\mathit {b0} \right ) {\mathrm e}^{-\frac {\arctan \left (\frac {\mathit {a1} x}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}}{\mathit {a1} \,x^{2}+\mathit {a0}}d x \right ) {\mathrm e}^{\frac {\arctan \left (\frac {\textit {\_b} \mathit {a1}}{\sqrt {\mathit {a0} \mathit {a1}}}\right )}{\sqrt {\mathit {a0} \mathit {a1}}}}+\textit {\_b} \mathit {c1} +\mathit {c0}}{\textit {\_b}^{2} \mathit {a1} +\mathit {a0}}d \textit {\_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 \(w(x,y)\)

\[ (a_1 x^2+a_0) w_x + (b_2 y^2+b_1 x y) w_y = (c_2 y^2+c_1 x^2) w + s_{22}y^2+s_{12} x y+s_{11} x^2+s_0 \]

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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