6.4.1 1.1

6.4.1.1 [1021] Example 1

problem number 1021

Added Feb. 17, 2019.

Chapter 4.1.1 example 1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a y w_y = b y^2 w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.1.2 [1022] Example 2

problem number 1022

Added Feb. 17, 2019.

Chapter 4.1.1 example 2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a y w_y = b e^{\lambda x} y w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-ax}} \right ) {{\rm e}^{{\frac {by{{\rm e}^{\lambda \,x}}}{a+\lambda }}}}\]

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6.4.1.3 [1023] Example 3

problem number 1023

Added Feb. 17, 2019.

Chapter 4.1.1 example 3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b w \]

Mathematica ✓

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

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

Maple ✓

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

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

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