Show, by means of the substitution $w={{x}^{2}}y$, that the differential equation $2y+x\frac{\mathrm

Question

Show, by means of the substitution $w={{x}^{2}}y$, that the differential equation $2y+x\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{5}{{{x}^{4}}}$ can be reduced to the form $\frac{\mathrm{d}w}{\mathrm{d}x}=-\frac{5}{{{x}^{3}}}$. [3] Hence, given that $y=3$ when $x=\frac{1}{2}$, solve the differential equation $2y+x\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{5}{{{x}^{4}}}$, to find $y$ in terms of $x$. [4]

Accepted Answer

(i) Shown (ii) $y = \frac{5}{2x^4} - \frac{37}{4x^2}$

2022 MI P1 Q2

2022 MI P1 Q2

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