# An equation of motion when a particle moves in a resting medium is given by dv/dt=−(kv+bt) where k and b are constants. Given that v=u when t=0, show that v(t)=b/k^2−b/kt+(u−b/k^2)e^(−kt)

An equation of motion when a particle moves in a resting medium is given by
$\frac{\mathrm{d}v}{\mathrm{d}t}=-\left(kv+bt\right)$
where k and b are constants. Given that v=u when t=0, show that
$v\left(t\right)=\frac{b}{{k}^{2}}-\frac{b}{k}t+\left(u-\frac{b}{{k}^{2}}\right){e}^{-kt}$
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Abigayle Lynn
${v}^{\prime }\left(t\right)=bt-kv\left(t\right)⟺$
$\frac{\text{d}v\left(t\right)}{\text{d}t}=bt-kv\left(t\right)⟺$
$\frac{\text{d}v\left(t\right)}{\text{d}t}+kv\left(t\right)=bt⟺$
Let

${e}^{kt}\cdot \frac{\text{d}v\left(t\right)}{\text{d}t}+\left({e}^{kt}k\right)v\left(t\right)={e}^{kt}bt⟺$
$\frac{\text{d}}{\text{d}t}\left({e}^{kt}v\left(t\right)\right)={e}^{kt}bt⟺$

${e}^{kt}v\left(t\right)=\frac{b{e}^{kt}\left(kt-1\right)}{{k}^{2}}+\text{C}⟺$
$v\left(t\right)=\frac{bt}{k}-\frac{b}{{k}^{2}}+\text{C}{e}^{-kt}$