7.3.1 Examples

7.3.1.1 [844] Example 1

problem number 844

Added Feb. 9, 2019.

Problem Chapter 3, example 1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

____________________________________________________________________________________

7.3.1.2 [845] Example 2

problem number 845

Added Feb. 9, 2019.

Problem Chapter 3, example 2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a e^x w_x + b w_y = c e^{2 x} y \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[x]*D[w[x, y], x] + b*D[w[x, y], y] == c*Exp[2*x]*y; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := a*exp(x)*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*exp(2*x)*y; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = -\frac {\left (-a \,{\mathrm e}^{-\frac {b \,{\mathrm e}^{-x}}{a}+x}+b \Ei \left (1, \frac {b \,{\mathrm e}^{-x}}{a}\right )\right ) c y \,{\mathrm e}^{\frac {b \,{\mathrm e}^{-x}}{a}}}{a^{2}}+\textit {\_F1} \left (y \,{\mathrm e}^{\frac {b \,{\mathrm e}^{-x}}{a}}\right )\]

____________________________________________________________________________________

7.3.1.3 [846] Example 3

problem number 846

Added Feb. 9, 2019.

Problem Chapter 3, example 3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a w_y = b \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == b; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\{\{w(x,y)\to b x+c_1(y-a x)\}\}\]

Maple ✓

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

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

____________________________________________________________________________________