Solving the Differential Equation (y'' - 4y e^{3t}) with Initial Conditions Using Laplace Transform

The Laplace transform is a powerful tool for solving differential equations, especially those with initial conditions. In this article, we will walk through the process of solving the differential equation y'' - 4y e^{3t} with the initial conditions y_0 0 and y'_0 0, using the Laplace transform.

Step 1: Take the Laplace Transform

First, we take the Laplace transform of both sides of the given equation. The Laplace transform of y(t) is denoted by Y(s). Using the properties of the Laplace transform, specifically:

[mathcal{L} {y(t)} Y(s) - y(0)]
[mathcal{L} {y'(t)} sY(s) - y(0)]

Given that y_0 0 and y'_0 0, we can simplify these to:

[mathcal{L} {y(t)} Y(s)]
[mathcal{L} {y'(t)} sY(s)]

Thus, taking the Laplace transform of the entire differential equation:

[s^2Y(s) - 4Y(s) mathcal{L} {e^{3t}}]

Step 2: Compute the Laplace Transform of e^{3t}

The Laplace transform of e^{3t} is well-known and can be derived as:

[mathcal{L} {e^{3t}} frac{1}{s - 3}]

Step 3: Substitute and Rearrange

Substitute the Laplace transform of e^{3t} into the transformed equation:

[s^2Y(s) - 4Y(s) frac{1}{s - 3}]

Rearranging this equation, we get:

[Y(s)(s^2 - 4) frac{1}{s - 3}]

Therefore, the expression for Y(s) is:

[Y(s) frac{1}{(s - 3)(s^2 - 4)}]

Step 4: Factor and Simplify

Factor the denominator s^2 - 4 as follows:

[s^2 - 4 (s - 2)(s 2)]

This leads to:

[Y(s) frac{1}{(s - 3)(s - 2)(s 2)}]

Step 5: Perform Partial Fraction Decomposition

To decompose the expression, let:

[Y(s) frac{A}{s - 3} frac{B}{s - 2} frac{C}{s 2}]

Multiplying through by the denominator, we obtain:

[1 A(s - 2)(s 2) B(s - 3)(s 2) C(s - 3)(s - 2)]

Substituting convenient values for s to solve for A, B, and C:

s 3: 1 A(1)(5) implies A frac{1}{5} s 2: 1 B(-1)(4) implies B -frac{1}{4} s -2: 1 C(-5)(-4) implies C frac{1}{20}

Thus, the partial fraction decomposition is:

[Y(s) frac{1}{5(s - 3)} - frac{1}{4(s - 2)} frac{1}{20(s 2)}]

Step 6: Take the Inverse Laplace Transform

The inverse Laplace transform of each term can be computed as follows:

(mathcal{L}^{-1} left{ frac{1}{5(s - 3)} right} frac{1}{5} e^{3t}) (mathcal{L}^{-1} left{- frac{1}{4(s - 2)}right} -frac{1}{4} e^{2t}) (mathcal{L}^{-1} left{ frac{1}{20(s 2)} right} frac{1}{20} e^{-2t})

Step 7: Combine the Results

Combining these results, the solution to the differential equation is:

[y(t) frac{1}{5} e^{3t} - frac{1}{4} e^{2t} frac{1}{20} e^{-2t}]

Final Answer

The solution to the differential equation is:

[y(t) frac{1}{5} e^{3t} - frac{1}{4} e^{2t} frac{1}{20} e^{-2t}]