Find a differential equation whose general solution is y=c_(1)e^(-2t)+c_(2) te^(-2t)

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2022-08-19

Find a differential equation whose general solution is y=c1e2t+c2te2t

Answer & Explanation

Adrienne Sherman

Adrienne Sherman

Beginner2022-08-20Added 9 answers

We want to find a differential equation whose general solution is
y=c1e2t+c2te2t...(1)
We know htat
y=c1eλ1t+c2teλ1t
is general solution of the equation
ay''+by' + cy=0
iff λ1 is the only root of the quadratic equation
aλ2+bλ+c=0...(2)
Therefore, we need to find coefficients a,b and c such that λ1=λ2=2 is the solution of the Eq. (2). The quadratic equation whose only root is -2 is
(λ+2)2=λ2+4λ+4
Therefore, one of the differential equations whose general solution is Eq. (1) is
y''+4y'+4=0
Result:
One example is y''+4y'+4=0.

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