Using the substitution $y = vx$, where $v$ is a function of $x$, solve \(x \frac{\mathrm{

Question

Using the substitution $y = vx$, where $v$ is a function of $x$, solve $x \frac{\mathrm{d}y}{\mathrm{d}x} - y = \sqrt{x^2 - y^2}$.

Accepted Answer

For $x>0$, $y=x\sin(\ln x+C_+)$; for $x<0$, $y=x\sin(-\ln(-x)+C_-)$, on intervals with nonnegative cosine. Also $y=\pm x$; compatible straight-line pieces may be joined at the sine-branch endpoints. At $x=0$, $y=0$ and differentiable matching are required.

Solving a homogeneous quation

Solving a homogeneous quation

Solution:

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