# Homework Help: Basic Initial Value Problems

1. Mar 10, 2009

### tracedinair

1. The problem statement, all variables and given/known data

Solve the following IVPs.

a) y'' + y = 0 for y(x0)=1 and y'(x0)=0

b) y'' - λ2y = 0 for y(x0)=0 and y'(x0)=1

2. Relevant equations
3. The attempt at a solution

a) Two distinct solutions for y'' + y = 0 are cos(x) and sin(x).

So, y(x0) = C1cos(x0) + C2sin(x0) = 1

y'(x0) = -C1sin(x0) + C2cos(x0) = 0

b) Two distinct solutions for y'' + λ2y = 0 is just e^($$\lambda$$x) and e^(-$$\lambda$$x)

So,

y(x0) = C1e^($$\lambda$$x0) + C2e^($$\lambda$$x0)) = 0

y'(x0) = C1λe^($$\lambda$$x0) - C2λe^(-$$\lambda$$x0) = 1

Now do I need to go through and solve for the C's or I am OK?

2. Mar 10, 2009

### tiny-tim

Yup!

3. Mar 10, 2009

### tracedinair

Thank you, my notes weren't very clear on the next step.