Question

0 <t<1 3. y' -5y = 10 t21 y(0)=1

Laplace transform

Answer 1

The given IvP is t² ostal 2 y'-sya iyoon=1 O tol t2 ostal where f(t) = o stol - convolution theorem : :- yi-5y = f(t) 1) = { Taking Laplace fronsform of the both sides of the above equation, we get 2{y'3-5 h{y} = [{f(t)} >> sx{} -760-5%{v} = {{tes >(6-5),{y} = {++){ +i (since yco)=1) {ys (-sjá {f(t)} tas -> jet)= a(as) & {f} **** +&+{*} If f(t) and get) be two functions then at {{{f(t)} {get)}} = f*g f* g is defined as fag = 5 + f(x) glt-x) dx *t { teas) ****esty Lite] S* fale54e) dx (gy correlation theorem) e 5(1-x) da where Now * {f(t)}} when out<l , Set femme =est *x15% da e-5x7t t ke -5K O

Š st žest test + - 5 25 -- viito se { **** } [") + ser]} -1) est (e-st_1) - t ist - East Bagest S tea) e 5(4-8) da s feny esetundets tes e 34-1) dx 25 125 1 when tl, 1 2 s ² esa est de dre est -56 ne da + Somezensed est? e**{***********] -5 O - e5t Site st-5 + -5 best [$€-(0)] 51-5 11 - 글 2_25t-5 25 125 + 12/25 (St-5) lisest - 37 125 e 2 125 est

5(1-x) When t=0, S. fcase dx =0 Therefore from (t) we get Sotest when t zo y ft) = 2 + 125 est test octal > --- 125 -37 ecst-5) St 125 + test tol 125 est t=0 i.e. 8+) = est t- 127 7125 obt21 2 -275-2/25 € 2/25 .-37 e(54-5)+ 127e5t 125 125 +7) ) This is the solution of the given IVP.