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:

enter image description here

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:

enter image description here

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Updated on March 27, 2020

Comments

  • axe
    axe over 3 years

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

    $v(t)$ is the velocity of the object

    Determine the unit impulse response AND the unit step response.

    How would i approach this question? do i set v(0 -) = 1 and solve or..?