Laplace transforms are particularly useful when solving ODEs with forcing terms that arediscontinuous. Let us solve the initial value problemy''+y=q(t),y(0)=1,y'(0)=0withq(t)={0,t<11,1t<20,t2This could be the equation of a spring that gets pushed with a constant force between timest=1 and t=2 and otherwise is left alone.(a) Derive the t-shift formula:IfH(t)is the Heaviside function, defined byH(t):={0,t<01,t0show thatL[H(t-a)f(t-a)]=e-asF(s)whereF(s)=L[f(t)].(b) Take the Laplace transform of both sides of ODE (2) and show that the Laplace transformofy(t)isY(s)=ss2+1+(e-s-e-2s)G(s)whereG(s)=1s(s2+1)(c) Take the inverse Laplace transform ofY(s) from (b)to find the solution y(t)of the initialvalue problem (2). You will have to use the t-shift formula from part (a),or rather theinverse t-shift formulaL-1[e-asF(s)]=H(t-a)f(t-a)