Solving RL Circuit Differential Equation via Laplace Transform

In this problem, we are asked to solve a first-order differential equation representing an R L circuit with a unit ramp input using the Laplace transform method.

Let's begin by taking the Laplace transform of both sides of the equation. We will denote the Laplace transform of small i of t as capital I of s.

Using linearity and the derivative property of the Laplace transform, the left side becomes R times I of s, plus L times the quantity s times I of s minus i of zero.

The Laplace transform of the ramp function t is one over s squared. We also substitute the initial condition, i of zero equals zero.

Now, we factor out I of s on the left side to begin isolating it.

Dividing both sides by the quantity L s plus R, we get our expression for I of s. To make calculation easier, let's factor L out of the denominator.

To find the inverse Laplace transform, we need to use partial fraction decomposition on this expression.

Multiplying through by the common denominator, we get one is equal to A times s times the quantity s plus R over L, plus B times the quantity s plus R over L, plus C times s squared.