Homogeneous linear equations 3

Calculus Level 1

What is the general solution to the differential equation y ′ ′ − 4 y ′ + 13 y = 0 y''-4y'+13y=0 ?

e 3 x ( c 1 cos ⁡ 3 x + c 2 sin ⁡ 3 x ) e^{3x}(c_1\cos 3x+c_2\sin 3x) e 2 x ( c 1 cos ⁡ 3 x + c 2 sin ⁡ 3 x ) e^{2x}(c_1\cos 3x+c_2\sin 3x) e 2 x ( c 1 cos ⁡ 2 x + c 2 sin ⁡ 3 x ) e^{2x}(c_1\cos 2x+c_2\sin 3x) e 3 x ( c 1 cos ⁡ 2 x + c 2 sin ⁡ 2 x ) e^{3x}(c_1\cos 2x+c_2\sin 2x)

Samir Khan by Samir Khan , 23, USA

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1 solution

Samir Khan
Jun 16, 2016

The characteristic equation is r 2 − 4 r + 14 = 0 ⟹ r = 2 ± 3 i r^2-4r+14=0\implies r=2\pm 3i , so the general solution is e 2 x ( c 1 cos ⁡ 3 x + c 2 sin ⁡ 3 x ) e^{2x}(c_1\cos 3x+c_2\sin 3x) .

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