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This is more of a mathematics question than an electronics question but what is the solution of this differential equation

$$α = \frac{V}{RC} + \frac{đV}{đt}$$

This solution gives the time response of the Ramp input in RC circuit. Can anyone show me the math behind it? The solution comes out to be

$$V = αRC(1 − e^{−t╱RC})$$

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Separate the variables and integrate: $$ α = \frac{V}{RC} + \frac{dV}{dt}$$ $$ \frac{dV}{dt} = α - \frac{V}{RC} $$ $$ (\frac{1}{α-\frac{V}{RC}})dV = dt $$ $$ RC\int \frac{dV}{αRC - V} = \int dt $$ $$ RC\ln(αRC - V).(-1) = t + C_1 $$ $$ \ln(αRC - V) = \frac{-t}{RC} + C_2 $$ $$ e^{-t/RC + C_2} = αRC - V $$ $$ e^{-t/RC}.C_3 = αRC - V $$ $$ V = αRC - e^{-t/RC}.C_3 $$ Assuming initial conditions to be V = 0 at t = 0,
$$ C_3 = αRC $$ $$ V = αRC - e^{-t/RC}.αRC $$ $$ V = αRC(1 - e^{-t/RC}) $$

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