# Classification of a Differential Equation relating multiple differentials

$$R_1C \frac{dv_o(t)}{dt} + v_o(t) = -R_2C \frac{dv_{in}(t)}{dt}$$

How should this Differential Equation be classified? It almost resembles the form of a Linear Differential Equation, but the differential on the right hand side leads me to believe otherwise.

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It is linear as long as $v_{in}'(t)$ does not depend nonlinearly on $v_0(t)$ or $v_0'(t)$. – Pragabhava Oct 30 '12 at 20:22
Okay, I think that makes sense (since $v'_{in}(t)$ could be a first order term). Would this then be solvable using the Method of Integrating Factors, or would I need to find another method? – Bradford Fisher Oct 30 '12 at 20:28
Method of integrating factor indeed. The thing is that, due the presence of the derivative, you'll be able to do an integration by parts. – Pragabhava Oct 30 '12 at 20:28
Awesome, thanks indeed! – Bradford Fisher Oct 30 '12 at 20:29

## 1 Answer

Another technique is a change of variables: Let $x = C R_1 v_{o}+C R_2 v_{in}$. Then $\dot{x} = C R_1 \dot{v_{o}}+C R_2 \dot{v_{in}}$, which gives $\dot{x} = -v_{o} = -\frac{x- C R_2 v_{in}}{C R_1} = -\frac{1}{C R_1} x+ \frac{R_2}{R_1} v_{in}.$ Solve for $x$, and recover $v_{o}$ with $v_{o} = \frac{x- C R_2 v_{in}}{C R_1}$.

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