# Consider the following initial value problem: y^N +49y= \left\{\begin{matrix} 7,\;0\leq t \leq 2...

## Question:

Consider the following initial value problem:

{eq}y^N +49y= \left\{\begin{matrix} 7,\;0\leq t \leq 2 & \\ 0, \;t>2\;\;\;\;\;& \end{matrix}\right. \hspace{10mm} y(0) = 3,\; y'(0) =0 {/eq}

Using {eq}Y {/eq} for the Laplace transformation of y(t), ie., {eq}Y= L { y(t) }, {/eq} find the equation you get by taking the Laplace transform of the differential equation and solve for {eq}Y(s) = \;\rule{20mm}{.5pt} {/eq}

## Initial Value Problem

There are many methods of solving initial value problems. When right hand side term of differential equation contains delta function or unit step function then it can be solve only by using Laplace transform. First we take Laplace transform, find an equation using initial conditions, then take inverse Laplace transform to find final solution.

{eq}L \left \{ f^n \right \}=s^nF(s)-s^{n-1}f(0)-s^{n-2}f^{'}(0)-\cdots -sf^{n-2}(0)-f^{n-1}(0) {/eq}

Apply inverse transform rule if {eq}L^{-1}\left\{F\left(s\right)\right\}=f\left(t\right)\mathrm{\:then}\:L^{-1}\left\{e^{-as}F\left(s\right)\right\}=u\left(t-a\right)f\left(t-a\right){/eq}

Where {eq}u(t){/eq} is unit step function

Consider the differential equation

{eq}{y}'' + 49y = \begin{cases} 7, \ \ 0\leq t\leq 2 & \\ 0, \ \ \ t\geq 2 & \end{cases} \cdots \cdots (1);\,...

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