(a) Consider the DE (Differential Equation) y+py+qy=r(t),y(0)=y(0)=0, where p,q are constants. Suppose that when r(t)=f(t) the solution of the DE is yf(t). Show that if r(t)=g(t) then the solution yg satisfies L{yg}=L{yf}L{f}L{g}. Deduce that you can use the response of the system to one input to predict the response to another input, without knowing the constants in the DE! (b) Consider a mechanical system described by my+y+ky=f(t),y(0)=0,y(0)=0. For a constant load f(t)=1,t0 the response of the system is observed to be y(t)=21(1et(sint+cost)). Suppose instead that the load applied is f(t)=e2t. Find the solution to the DE giving the system response in this scenario. (a) Consider the DE (Differential Equation) y+py+qy=r(t),y(0)=y(0)=0, where p,q are constants. Suppose that when r(t)=f(t) the solution of the DE is yf(t). Show that if r(t)=g(t) then the solution yg satisfies L{yg}=L{yf}L{f}L{g}. Deduce that you can use the response of the system to one input to predict the response to another input, without knowing the constants in the DE! (b) Consider a mechanical system described by my+y+ky=f(t),y(0)=0,y(0)=0. For a constant load f(t)=1,t0 the response of the system is observed to be y(t)=21(1et(sint+cost)). Suppose instead that the load applied is f(t)=e2t. Find the solution to the DE giving the system response in this scenario