Find the solution of the given initial value problem. y'−2y=e2t,y(0)=2

facas9 2021-10-21 Answered
Find the solution of the given initial value problem. y2y=e2t,y(0)=2
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Expert Answer

Usamah Prosser
Answered 2021-10-22 Author has 86 answers
Write equation in the form of standard Linear differential equation
dydt+P(t)y=Q(t)
Compare with it and get P(t).
Now calculate integrating factor μ=eP(t)dt
Multiply each term of differential equation by integrating factor
P(t)=2
Integrating factor =e(2)dt
μ=e2t
Multiply equation by integrating Factor we get,
e2ty2e2ty=e2te2t
e2ty2e2ty=1
Consider (e2ty)
Differentiate with respect to t we get,
(e2ty)=e2ty2e2ty
Plug this value in equation
we get,
(e2ty)=1
Integrate both sides,
(e2ty)=1
e2ty=t+C
Where C is integration constant
Apply initial condition. Plug t=0 in equation
e0y(0)=0+C
y(0)=C
Given y(0)=2
C=2
Plug value of C=2 in equation
e2ty=t+2
Multiply e2t both sides we get,
y=e2t(t+2)

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