Question

(vi) $\sqrt{a+x\frac{dy}{dx}}+x=0$

Answer 1

y = \frac{x^2}{2} - a \ln|x| + C, where x < 0

Answer 2

The differential equation is first rearranged to isolate the square root. The condition for the square root to be real ($-x \ge 0 \implies x \le 0$) is noted. Squaring both sides transforms the equation into a separable first-order differential equation. Variables are separated, and both sides are integrated. The constant of integration is added. Finally, the domain restriction $x < 0$ (due to $\ln|x|$ and the initial condition) is applied, leading to the general solution.