It is given that ${{x}^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}-3xy+4=0$. Using the substitution $y=u{{x}

Question

It is given that ${{x}^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}-3xy+4=0$. Using the substitution $y=u{{x}^{3}}$, show that the differential equation can be transformed to $\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{c}{{{x}^{5}}}$, where $c$ is a constant to be determined. [3] Hence given that $y=3$ when $x=1$, solve the differential equation ${{x}^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}-3xy+4=0$ to find $y$in terms of $x$. [4]

Accepted Answer

(i) $\frac{\mathrm{d}u}{\mathrm{d}x} = -\frac{4}{x^5}$; (ii) $y = \frac{1}{x} + 2x^3$

2021 MI P1 Q3

2021 MI P1 Q3

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