# I have a first order linear differential equation (a variation on a draining mixing tank problem) wi

I have a first order linear differential equation (a variation on a draining mixing tank problem) with many constants, and want to separate variables to solve it.
$\frac{dy}{dt}={k}_{1}+{k}_{2}\frac{y}{{k}_{3}+{k}_{4}t}$
y is the amount of mass in the tank at time t, and for simplicity, I've reduced various terms to constants, ${k}_{1}$ through ${k}_{4}$.
Separation of variables is made difficult by ${k}_{1}$, and I've considered an integrating factor, but think I might be missing something simple.
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Alisa Jacobs
$\frac{dy}{dt}={k}_{1}+{k}_{2}\frac{y}{{k}_{3}+{k}_{4}t}$
For simplicity substitute $u={k}_{3}+{k}_{4}t$
${k}_{4}\frac{dy}{du}={k}_{1}+{k}_{2}\frac{y}{u}$
${y}^{\prime }-\frac{{k}_{2}}{{k}_{4}u}y=\frac{{k}_{1}}{{k}_{4}}$
Solve the homogeneous DE and use variation of parameter method for the inhomogeneous DE:
${y}^{\prime }-\frac{{k}_{2}}{{k}_{4}u}y=0$

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