📘 laplace transform

1. The problem is to find the inverse Laplace transform of the function $$\frac{12s}{(s-5)(s+1)}$$. 2. The formula used is the partial fraction decomposition for rational functions

1. **Problem:** Find the Laplace Transform of $$F(t) = \frac{e^{t - \cos t}}{t}$$. 2. **Recall the Laplace Transform definition:**

1. **Problem Statement:** Find the Laplace Transform of the function $$f(t) = \begin{cases} \frac{t}{k}, & 0 < t < k \\ 1, & t > k \end{cases}$$ where $k$ is a positive constant. 2

1. **State the problem:** Find the inverse Laplace transform of $$\frac{5}{s^2(c s + 4)}$$ using the convolution theorem and then perform division by $s$. 2. **Recall the convoluti

1. **State the problem:** Find the inverse Laplace transform of $$F(s) = \frac{(s - 2) e^{-2s}}{s^2 - 6s + 13}$$. 2. **Recall the shifting theorem:** The factor $$e^{-as}$$ in the

1. Problema: Determinați transformata Laplace a funcției $f(t) = te^{3t}$ pentru $t \geq 0$ și $f(t) = 0$ pentru $t < 0$ folosind teorema derivării imaginii. 2. Formula folosită: D

1. **State the problem:** Find the inverse Laplace transform of $$\frac{s+5}{s^2 - 2s - 3}$$. 2. **Factor the denominator:** The quadratic $$s^2 - 2s - 3$$ factors as $$ (s - 3)(s

1. **State the problem:** Find the inverse Laplace transform of the function $$F(s) = \frac{s + 5}{s^2 - 2s - 3}$$. 2. **Factor the denominator:** The quadratic in the denominator

1. **Problem statement:** Find the inverse Laplace transform of the given functions: (a) $$F(s) = \frac{7}{(s+3)^3}$$

1. **State the problem:** Find the inverse Laplace transform of the function $$F(s) = \frac{e^{-(s+1)}}{(s+1)(s^2 + 2s + 10)}.$$\n\n2. **Recall the shifting theorem:** The factor $

1. **Problem statement:** Find the inverse Laplace transform of the function given by the infinite series $$F(s) = \sum_{m=0}^\infty \frac{\Gamma(m+1)}{\Gamma(\alpha m + \beta)} s^

1. **State the problem:** We want to find the inverse Laplace transform of the function given by the infinite sum:

1. **State the problem:** We want to find the inverse Laplace transform of the function given by the infinite sum $$F(s) = \sum_{m=0}^\infty \frac{\Gamma(m+1)}{\Gamma(\alpha m + \b

1. **State the problem:** Find the inverse Laplace transform of the function $$F(s) = \frac{2s - 3}{s^2 - 4}$$. 2. **Factor the denominator:** Note that $$s^2 - 4 = (s - 2)(s + 2)$

1. **Problem:** Given that $L\{t\sin(\omega t)\} = \frac{2\omega s}{(s^2+\omega^2)^2}$, find $L\{\omega t \cos(\omega t) + \sin(\omega t)\}$. 2. **Approach:** We use linearity of L