It is given that $2xy - x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = 2x^4 + y^2$. Using the substitu

Question

It is given that $2xy - x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = 2x^4 + y^2$. Using the substitution $wy = x^2$, show that the differential equation can be transformed to $\frac{\mathrm{d}w}{\mathrm{d}x} = 1 + 2w^2$.[3] Hence, solve the differential equation $2xy - x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = 2x^4 + y^2$.[3] It is given that $u$ satisfies $\frac{\mathrm{d}^2u}{\mathrm{d}x^2} = e^{2x}$ and has a stationary value of $\frac{e^2}{4}$ when $x = 1$. Find $u$ in terms of $x$.[3]

Accepted Answer

(a)(i) $\frac{dw}{dx}=1+2w^2$ (a)(ii) $y=\frac{\sqrt2x^2}{\tan(\sqrt2(x+C))}$ (b) $u=\frac14e^{2x}-\frac12e^2x+\frac12e^2$

2024 EJC P1 Q5

2024 EJC P1 Q5

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