The given ODE is
[tex] \frac{dy}{dx} = \frac{x}{y} [/tex]
Let y² = x - v
Then
[tex]2y \frac{dy}{dx} =1 - \frac{dv}{dx} \\\\ \frac{dy}{dx} = \frac{1}{2y} (1- \frac{dv}{dx} )[/tex]
Substitute in the original ODE.
[tex] \frac{1}{2y} (1- \frac{dv}{dx}) = \frac{x}{y} \\\\ 1- \frac{dv}{dx} =2x \\\\ \frac{dv}{dx} =1-2x [/tex]
Integrate with respect to x to obtain
v = x - x² + b
Hence obtain
y² = x² + b
When y(x) passes through (0,0), obtain
b = 0
y² = x²
When y(x) passes through (1,0), obtain
0 = 1 + b => b = -1
y² = x² - 1
When y(x) passes through (0,1), obtain
1 = 0 + b => b = 1
y² = x² + 1
A graph of the three solutions is shown below.
