Use the Laplace transform to solve the given initial-value problem.
$$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\{\begin{array}{rr} 0, & 0 \leq t<1 \\ 5, & t \geq 1 \end{array}\right. $$
Use Definition 7.1.1.DEFINITION 7.1.1 Laplace TransformLet \(f\) be a function defined for \(t \geq 0\). Then the integral$$ \mathscr{L}\{f(t)\}=\int_{0}^{\infty} e^{-s t} f(t) d t $$is said to be the Laplace transform of \(f\), provided that the integral converges.Find \(\mathscr{L}\{f(t)\}\). (Write your answer as a function of s.)\(f(t)=\left\{\begin{array}{lr}t, & 0 \leq t<1 \\ 1, & t \geq 1\end{array}\right.\)
Use the Laplace transform to solve the given initial-value problem. so, 0 <t< 1 y' + y = f(t), y(0) = 0, where f(t) 17, t21 y(t) = + ult-
Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y" + 25y = f(t), y(0) = 0, y (O) = 1, where RE) = {cos(5€), Ostan (Σπ rce) = f sin(51) + (t-1) -sin 5(t-T) 5 Jault- TE ) X
Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y" + y = f(t), y(0) - 1, 0) = 0, where - (1, osta 1/2 f(0) = sin(t), t2/2 . 70 y() = 1 (4- 7 )sin(e- 1 + cost- -cos( - ) Dale X Need Help? Read Watch Talk to a Tutor Submit Answer
(25 marks) Solve the following initial value problem using Fourier transform.$$ \begin{array}{l} u_{t}=u_{x x}, \quad-\infty< x <\infty, t= >0 \\ u(x, 0)=\left(1-2 x^{2}\right) e^{-4 x^{2}}, \quad-\infty< x <\infty \end{array} $$with \(u(x, t) \rightarrow 0\) and \(u_{x}(x, t) \rightarrow 0\) as \(x \rightarrow \pm \infty\).
Use the Laplace transform to solve the given initial-value problem. 0 st<1 t 1 y' y f(t), y(0) 0, where f(t) (4, ae-1 -(1-1) 4 y(t) X Use the Laplace transform to solve the given initial-value problem. 0 st
QUESTION 3 Use Laplace Transform to solve the initial value problem y" + 9y = f(t) ,y(0) = 1, y'(0) = 3 where 6, f(t) 0 <t<nt i < t < 0
Use the Laplace transform to solve the given initial value problem. y" – y' – 12y = 0; y(0) = 1, y'(0) = -1 (t) =
Use the Laplace transform to solve the given initial-value problem. y" – 3y' = 8e2t – 2e-, y(0) = 1, y'(0) = -1 y(t) =
Use the Laplace transform to solve the given initial-value problem. y" + 6y' + 5y = 0, y(0) = 1, y'(O) = 0 y(t) =