Question

# Obtain the Laplace Transform of Lleft{e^{-2x}+4e^{-3x}right}

Laplace transform
Obtain the Laplace Transform of $$L\left\{e^{-2x}+4e^{-3x}\right\}$$

2021-03-19
Step 1
Laplace Transform: The Laplace transform provides an effective tool or way of solving initial-value problems for linear differential equations with constant coefficients. The type of problem differs accordingly with the question.
Step 2
The formula for calculating Laplace transform of $$e^{ax}=\frac{1}{(s-a)}$$
According to the question: $$L(e^{-2x} +4e^{-3x})$$ we can separate the two terms
$$L(e^{-2x})=\frac{1}{(s+2)}$$
$$L(e^{-3x})=\frac{1}{(s+3)}$$
$$=\frac{1}{(s+2)}+\frac{4}{(s+3)}$$
$$=\frac{(s+3+4s+8)}{(s+2)(s+3)}$$
$$=\frac{(5s+11)}{(s+2)(s+3)}$$
$$=\frac{(5s+11)}{(s^2+3s+2s+6)}$$
$$=\frac{(5s+11)}{(s^2+5s+6)}$$ is the answer.