how do you find analytical solution of ode y’+2y=e^(-4t)+cos(2pit) such that y(0)=1

Question : Find analytical solution of ode \[y’+2y=e^(-4t)+cos(2pit)\] such that y(0)=1Solution :Given differential equation\[y^\prime+2y=e^{-4t}+\cos{\left(2\pi t\right)},y\left(0\right)=1\] (1)This equation can be written in the form \[y^\prime+P\left(t\right)y=Q\left(t\right)\]Which is Linear ODE of first order whose general solution is written as\[y\left(t\right)u\left(t\right)=\int Q\left(t\right)u\left(t\right)dt+c \]Where u(t) is…

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