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in Differential Equations by (120 points)
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Solve the differential equation \[ \frac{d^{2} x}{d t^{2}}-4 \frac{d x}{d t}+13 x=0 \text { with } x(0)=0, \frac{d x}{d t}(0)=2 \]

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1 Answer

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by (45.0k points)

Given differential equation is

\(\frac{d^2x}{dt^2}-4\frac{dx}{dt}+13x=0\)

It's auxiliary equation is

m2 - 4m + 13 = 0

⇒ m = \(\frac{4\pm\sqrt{16-52}}2\) = \(\frac{4\pm 6i}2\) = 2\(\pm\)3i

∴ x = e2t(C1 cos 3t + C2 sin 3t)

\(\frac{dx}{dt} \) = e2t(2C1 cos 3t + 2C2 sin 3t - 3C1 sin 3t + 3C2 cos 3t)

 = e2t((2C1 + 3C2) cos 3t + (2C2 - 3C1) sin 3t)

Given that x(0) = 0

∴ C1 = 0

also given that \(\frac{dx}{dt}(0)=2\)

∴ 2C2 = 2

⇒ C2 = 2/3

∴ x = e2t(2/3 sin 3t)

⇒ 2/3 e2t sin 3t which is solution of given differential equation.

by (120 points)
Thank you
Im getting answer as x = e²t (2cos3t -2/3 sin3t ) . Is it correct ?

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