Answer to Question #231465 in Differential Equations for John

Question

Find the complete integral of p+q=x+y

Answer 1

Solution

Let’s consider the transformation  α = x – y,  β = y

Inverse transformation is x = α + β,  y = β

Let u(x, y) = v(α, β). Then, we have

p + q = u_x + u_y =  v_α (∂α/∂x) + v_β (∂β /∂x) + v_α (∂α/∂y)  + v_β (∂β /∂y)  =  v_α - v_α + v_β = v_β 

 

Therefore, the partial differential equation becomes

v_β =  α + 2β

Integrating this equation

v = α β + β² + g(α) 

Here g(α) is an arbitrary function of α.

So u(x, y) = α β + β² + g(α) = (x – y)y+y² + g(x – y) = x y + g(x – y)

Answer

u(x, y) = x y + g(x – y), where g(α) is an arbitrary function of α.