Solve the differential equation \frac{d^2y}{dx^2}-2\frac{dy}{dx}+4y=e^x\sin^2(\frac{x}{2})

sputavanomr 2021-11-15 Answered
Solve the differential equation
d2ydx22dydx+4y=exsin2(x2)
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Expert Answer

Mary Ramirez
Answered 2021-11-16 Author has 19 answers
The general solution will be the sum of the complementary solution and particular solution.
The complementary solution for the given equation:
d2ydx22dydx+4y=exsin2(x2)
λ22λ+4=0
λ=b±b24ac2a
b+b24ac2a=(2)+(2)24(1)(4)2(1)=1+3i
bb24ac2a=(2)(2)24(1)(4)2(1)=13i
yp(x)=ex(c1cos(3x)+c2sin(3x))
The particular for the equation is:
exsin2(x2)D22D+4=ex(1cosx2)D22D+4=12ex(1cosx)D22D+4
=12exD22D+412excosxD22D+4
P.I=1f(D)eax=1f(a)eax
exD22D+4=ex12(1)+4=ex
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