Solve the differential equation?

Question

Solve the differential equation y^(4)-5y"-36y=0.

r^4-5r^2-36

(r^2-9)(r^2+4)

r=3, -3, 2i, -2i

y=c1*e^3x+c2*e^-3x+c3(cos(2x))+c4(sin(2x))

Is that the right answer?

And can someone give me some websites that has lots of practice problems like this for differential equations?

Answer 1

Let's verify it.

You got the characteristic polynomial solved correctly. It indeed factors and has the four roots you listed.

Using the roots we can make a basis.

The initial basis:

{e^3x, e^-3x, e^2ix, e^-2ix}

However, the last two solutions involve complex numbers so we must replace them.

e^2ix = cos 2x + i sin 2x

e^-2ix = cos 2x - i sin 2x

(e^2ix + e^-2ix) / 2 = cos 2x

(e^2ix - e^-2ix) / 2i = sin 2x

Hence the updated basis is {e^3x, e^-3x, cos 2x, sin 2x}

Every solution must be a linear combination of these four so you get the following.

y = C₁ e^3x + C₂ e^-3x + C₃ cos 2x + C₄ sin 2x

Thus you are correct.