Solving Initial Value Problems for Second Order Differential Equations

[tracedinair]

Homework Statement

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

Homework Equations

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?

 

[tiny-tim]

Now do I need to go through and solve for the C's … ?

Yup!

 

[tracedinair]

Yup!

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