Solution to the Differential Equation Using Laplace Transforms

The given differential equation can be solved using Laplace transform. Laplace transform is used to solve linear differential equations with constant coefficients. The equation can be written as:

5 dy dt + 4y = 2 y(0) = 1

Step One: Taking Laplace Transforms of Both Sides

The first step is to take the Laplace transform of both sides of the equation. The Laplace transform of the equation is:

5 sL{y} + 4L{y} = 2L{1}

Step Two: Solving for the Laplace Transform of y

The Laplace transform of y can be solved using algebra. The equation can be written as follows:

L{y} = (2L{1} – 5s) / 4 = (2 – 5s) / 4

Step Three: Taking the Inverse Laplace Transform

The last step is to take the inverse Laplace transform of the equation. The inverse Laplace transform of the equation is:

y(t) = (2e5t – 1) / 4

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