Show, by using the substitution $w=\frac{y}{{{t}^{2}}}$, that the general solution of the differenti

Question

Show, by using the substitution $w=\frac{y}{{{t}^{2}}}$, that the general solution of the differential equation $t\frac{\mathrm{d}w}{\mathrm{d}t}={{w}^{2}}{{t}^{3}}+2wt-2w$ is $w=\frac{1}{{{t}^{2}}}\left( \frac{2A{{e}^{2t}}}{1-A{{e}^{2t}}} \right)$. [5] Find the particular solution, given that $w=\frac{2{{e}^{2}}}{1-{{e}^{2}}}$ when $t=1$ . Sketch the particular solution curve. [2]

Accepted Answer

(i) $w=\frac{1}{t^2}\left(\frac{2Ae^{2t}}{1-Ae^{2t}}\right)$. (ii) $A=1$, so $w=\frac{2e^{2t}}{t^2(1-e^{2t})}$, with asymptotes $t=0$ and $w=0$.

2010 ACJC P1 Q12

2010 ACJC P1 Q12

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