Question

Homogeneous constant coefficients

y''-2y'+2y=0

Initial conditions y(0)=0
y'(0)=-5

I don't know where to start! I think I need to use imaginary numbers??

Answer 1

Let y = exp(ax). Substitute this into your equation.

You will get a polynominal in a equal to zero. Solve that polynominal and hence find two homogeneous solutions y1 and y2.

The general solution is y = c1.y1 + c2.y2

Substitute now your initial conditions and solve for c1 and c2.

If this is utterly mysterious to you, watch this lecture:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-9-solving-second-order-linear-odes-with-constant-coefficients/

Answer 2

for clarification: I use the exp(ax) with a as the constant? Just substitute exp(ax) into the original EQ and apply whatever power to it that the prime is? Example y''= exp(ax)^2

Answer 3

No. y'' is the second derivative. And yes, a is a constant.

Answer 4

I think you should almost certainly watch the lecture. You'll learn a lot.