Use the substitution $y=vx$ to show that the differential equation \[x^2\frac{\mathrm{d}y}{\mathrm

Question

Use the substitution $y=vx$ to show that the differential equation $$x^2\frac{\mathrm{d}y}{\mathrm{d}x}=y^2+xy$$ reduces to $$x\frac{\mathrm{d}v}{\mathrm{d}x}=v^2.$$[3] Hence find the general solution of the differential equation $$x^2\frac{\mathrm{d}y}{\mathrm{d}x}=y^2+xy.$$ Give your answer in the form $y=\mathrm{f}(x)$, where $\mathrm{f}(x)$ is a function to be determined.[3]

Accepted Answer

(a) Shown (b) $y=-\dfrac{x}{\ln\left|x\right|+C}$, where $C$ is an arbitrary constant

N2026 P1 Q4

N2026 P1 Q4