2020 RVHS P2 Q2
A particle moving in a liquid is such that after $t$ seconds, its velocity is $v$ ms$^{-1}$, $v\ne 0$. $v$ satisfies the differential equation $v\frac{\text{d}v}{\text{d}t}+2{{v}^{2}}=t{{\text{e}}^{-2t}}$.
(i)
By using the substitution $y=v{{\text{e}}^{2t}}$, show that $y\frac{\text{d}y}{\text{d}t}=t{{\text{e}}^{2t}}$.
[3]
(ii)
Hence find ${{v}^{2}}$ in terms of $t$ where $\frac{\text{d}v}{\text{d}t}=-8$ when $t=0$.
[6]
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