6.3.1 Examples

6.3.1.1 [843] Example 1

problem number 843

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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\frac {1}{a} \left ( \ln \left ( x \right ) c+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) a \right ) }\]

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6.3.1.2 [844] Example 2

problem number 844

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]];
 

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

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'));
 

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

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6.3.1.3 [845] Example 3

problem number 845

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]];
 

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

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'));
 

\[w \left ( x,y \right ) =bx+{\it \_F1} \left ( -ax+y \right ) \]

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