Question 4 only. Please explain your work.

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【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/3Solution:The given differential equation is{dy}/{dt} = ky-e^{t}y(0) =y_{0}, To solve the given differential equation, we will use the integrating factor method. We have, M(t) = e^{-∫k dt} = e^{-kt} Multiplying both sides of the given differential equation by M(t), we get e^{-kt} {dy}/{dt} = kye^{-kt} - e^{t-kt} Integrating both sides, we get ∫ e^{-kt} {dy}/{dt} dt = ∫kye^{-kt} dt - ∫e^{t-kt} dt On the left side, we have e^{-kt} y = ∫ e^{-kt} {dy}/{dt} dtOn the right side, we have ∫kye^{-kt} dt = kye^{-kt} and ∫e^{t-kt} dt = e^{t-kt} + c where c is the constant of integration. Explanation:Please refer to solution in this step.Step2/3Thus, we havee^{-kt} y = kye^{-kt} + e^{t-kt} + cRearranging the above equation, we gety = (ky+1)e^{t-kt} + c e^{kt}Substituting the initial condition y(0) = y0, we gety0 = (ky+1) + ce^{k0}orc = y0 - (ky + 1)Therefore, the solution of the given differential equation isy = (ky+1)e^{t-kt} + (y0 - (ky + 1))e^{kt}Now ... See the full answer