How to solve $\frac{dV}{dt} = k_1 \cdot h \cdot \frac{dh}{dt} + k_2 \cdot h^2$

Question

How can one solve the following equation?

$$\frac{dV}{dt} = k_1 \cdot h \cdot \frac{dh}{dt} + k_2 \cdot h^2$$

I got to it by modeling volume decrease using related rates, but unlike a typical cone-shaped-melting-icecream, I have a shape with a cross section that varies.

I am assuming I need to be able to solve a differential equation, correct? If so, what type and technique should I use?

Any guidance is highly appreciated.

Answer 1

The equation is:

$$\frac{dV}{dt}=\frac{k_1}2\frac d{dt}(h^2)+k_2h^2.$$

Changing variables, i.e., letting $W=\frac2{k_1}V$, $H=h^2$, and $c=\frac{k_2}{2k_1}$, we obtain:

$$\frac{dW}{dt}=\frac{dH}{dt}+cH,$$ for some constant $c$. Further simplifications are not possible without additional information.