Question

Consider the equation yu_x - xu_y = 0, (y>0). Check for each of the following initial conditions whether the problem is solvable. If it is solvable, find a solution. If it is not, explain why:

a) u(x,0) = x^2
b) u(x,0) = x
c) u(x,0) = x, x>0

Answer 1

Conditions

Consider the equation $yu\_x - xu\_y = 0$, $(y\>gt;0)$. Check for each of the following initial conditions whether the problem is solvable. If it is solvable, find a solution. If it is not, explain why:

a) $u(x,0) = x^2$

b) $u(x,0) = x$

c) $u(x,0) = x, x\>gt;0$

Solution

$$y \frac{du}{dx} - x \frac{du}{dy} = 0$$ $$\frac{dx}{y} = \frac{dy}{x} = \frac{du}{0}$$ $$\varphi_1(x,y,u) = u$$ $$\frac{dx}{y} = \frac{dy}{x}$$ $$x^2 - y^2 = c$$ $$\Phi(u, x^2 - y^2) = 0$$ $$u = f(x^2 - y^2)$$

Where $f$ has a derivative at some interval.

So we can see now, that the problem is solvable for $1^{st}$ condition.