Find the Green's function G for the initial value problem x^(')+x=f(t);x(0)=0 and use it find the solution of the initial value problem in the case f(t)=t. Bonus. Verify directly that x(t)=int_0^t G(t- au )f( au )d au satisfies the given initial value problem for any f. Find the Green's function G for the initial value problem x'+ x = f(t); x(0)= 0 and use it find the solution of the initial value problem in the case f (t) = t. Bonus. Verify directly that x(t) = (' G(t -- T)f (T) dt satisfies the given initial value problem for any f .
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To find the Green's function G, we first solve the homogeneous equation x'+ x = 0. The solution to this equation is x(t) = Ae^(-t), where A is a constant determined by the initial condition x(0) = 0. Since x(0) = A = 0, the solution to the homogeneous equation is Show more…
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