(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve the following IVPs.

a) y'' + y = 0 for y(x_{0})=1 and y'(x_{0})=0

b) y'' - λ^{2}y = 0 for y(x_{0})=0 and y'(x_{0})=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(x_{0}) = C_{1}cos(x_{0}) + C_{2}sin(x_{0}) = 1

y'(x_{0}) = -C_{1}sin(x_{0}) + C_{2}cos(x_{0}) = 0

b) Two distinct solutions for y'' + λ^{2}y = 0 is just e^([tex]\lambda[/tex]x) and e^(-[tex]\lambda[/tex]x)

So,

y(x_{0}) = C_{1}e^([tex]\lambda[/tex]x_{0}) + C_{2}e^([tex]\lambda[/tex]x_{0})) = 0

y'(x_{0}) = C_{1}λe^([tex]\lambda[/tex]x_{0}) - C_{2}λe^(-[tex]\lambda[/tex]x_{0}) = 1

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

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# Basic Initial Value Problems

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