# Unit impulse / step response of a 1st order differential equation

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We are given:

$$10v'(t) + 0.6 v(t) = f(t)$$

Since the impulse input is the derivative of the unit step input, the impulse response of a linear system is the derivative of its unit step response. Hence, in order to get the impulse response, first obtain the unit step response, and then differentiate the unit step response with respect to time. This procedure is easier.

As a first step, we set $f(t) = u(t)$, the Heaviside unit step function, and solve:

$$10 v'(t) + \dfrac{6}{10} v(t) = u(t), ~~v(0-) = 0$$

This results in:

$$v_{unit}(t) = \dfrac{5}{3} e^{-\frac{3 t}{50}} \left(e^{\frac{3 t}{50}}-1\right) u(t)$$

A plot of the unit response is:

Next, we want to find the impulse (Dirac delta function) response by taking the derivative of $v_{unit}(t)$. This results in:

$$v_{impulse}(t) = \begin{cases}\dfrac{1}{10}e^{-\frac{3t}{50}}, & t > 0, \\ \infty, & t = 0.\end{cases}$$

A plot of the impulse response is:

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Updated on March 27, 2020

You are given the equation $10v'(t) + 0.6 v(t) = f(t)$
$v(t)$ is the velocity of the object