Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.

Question

Accepted Answer

A. $y + \sqrt{x^2 + y^2} = cx^2$

**Homogeneous:** put $y = vx$.
**After substitution:** $x\,dv/dx = \sqrt{1+v^2}$
⇒ $dv/\sqrt{1+v^2} = dx/x$.
**Integrating:** $\log(v + \sqrt{1+v^2}) = \log(cx)$
⇒ $v + \sqrt{1+v^2} = cx$.
Back-substitute $v = y/x$: $y/x + \sqrt{1 + y^2/x^2} = cx$, which simplifies to $y + \sqrt{x^2+y^2} = cx^2$.

Answer

$y^2 = cx$

Answer

$\sqrt{x^2 + y^2} = c$

Answer

$y = x + c$

Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.

Keep practising Unit 7: Differential Equations and Modeling