Sketch the function given below. f(t) = sin (omega 0 t) u(t). Using the definiti

Question

Sketch the function given below. f(t) = sin (omega 0 t) u(t). Using the definition of the Laplace transform (not tables), go through all the calculus and algebra to calculate the transform of the above equation. Let f(t) F(s) and x(t) X(s) Find F(s) in terms of X(s) and f(t) in the following two cases: (You do not have to find f(t) that satisfies the differential equation.) x(t) = a d / dt f(t) + bf(t) x(t) = a d2 / dt2 f(t) + b d / dt f(t) + cf(t).

Explanation / Answer

Q.3) in laplace derivative is equivalent to s i mean if X(t)-> X(S) then dX/dt=s*x so using this formula lets solve the 4th question x(t)=a* df/dt + b*f(t) apply laplace on both sides you then get X(S)=a*s*(f(S))+b*(f(S)) X(S)=(as+b)*(f(S)) so (f(S))= X(S)/(as+b) lets now solve 5th que x(t)=a* d^2f/dt^2 + b*df(t)/dt + c* (f(t)) just like above problem apply laplace on both sides you then get X(S)=a*s*s*(f(S))+b*s*(f(S))+c*(f(S)) which means (f(S))= X(S)/(as^2+bs+c)

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