MMfE - Solve Differential Equations Using the Laplace Transformation

All Exercises (with Solutions) containing the “solve_differential_equation_using_Laplace_transform” tag:

• MMfE ~ exam_2020_06_10 - with solutions
• MMfE ~ exam_2020_06_24 - with solutions
• MMfE ~ exam_2020_07_15 - with solutions
• MMfE ~ exam_2021_06_25 - with solutions
• MMfE ~ exam_2021_09_10 - with solutions

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General Solution

1. Use the Laplace Transforms of: $\mathcal{L}\left[\frac{d^n}{d{t}^n}f(t)\right]=s^n\cdot F(s)+s^{n-1}\cdot f(0)+s^{n-2}\cdot f^{\prime }(0)+s^{n-3}\cdot f^{\prime \prime }(0)+\dots +f^{(n)}(0)$
2. Then after finding the equation of $F(s)=\dots$ use the Residue theorem to find the Inverse Laplace Transform.

• The Inverse Laplace Transform is given by:

$$f(t)={\int }_{\sigma -i\infty }^{\sigma +i\infty }F(s)e^{st}ds$$

Where $\sigma$ is the abscissa of convergence

• Then applying the Jordan’s Lemma:

• Here is a recap of the Residue Theorem

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~Ex.: 2020_06_10 Point 4

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~Ex.: 2020_06_24 Point 4.

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~Ex. 2020_07_15 Point 4.

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~Ex.: 2021_06_25 Point 4.

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~Ex.: 2021_09_10 Point 4.

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