Answer with Step-by-step explanation:
We are given that
DE
[tex]x''+x=2e^t[/tex]
Function:[tex]x=C_1sint+C_2cost+e^t[/tex]
We have to show that given function is a solution of the equation for all values of the constants.
If given function is solution of DE then it satisfied the given DE.
Differentiate function w.r.t.t
[tex]x'=C_1cost-C_2sint+e^t[/tex]
Again differentiate w.r.t. t
[tex]x''=-C_1sint-C_2cost+e^t[/tex]
Substitute the values in the given DE
[tex]-C_1sint-C_2cost+e^t+C_1sint+C_2cost+e^t=2e^t[/tex]
LHS=RHS
Given function satisfied the given DE.Therefore, it is solution of given DE for all values of the constants.
