allerySolve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.
A$y + \sqrt{x^2 + y^2} = cx^2$
B$y^2 = cx$
C$\sqrt{x^2 + y^2} = c$
D$y = x + c$
Answer & Solution
Correct 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 $y + \sqrt{x^2+y^2} = cx^2$.
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