Question 1
Solve the third order differential equation:
- none of these answers
Explanation: So this is a homogenous, third order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:
Which, using the cubic formula or factoring gives us roots of 
, 
and 
The solution of homogenous equations is written in the form:
so we don't know the constants, but can substitute the values we solved for the roots:
We have three initial values, one for y(t), one for y'(t), and for y''(t) all with t=0
So:
so: 
So this can be solved either by substitution or by setting up a 3X3 matrix and reducing. Once you do either of these methods, the values for the constants will be: 
Then 
and 
This gives a final answer of:
Which, using the cubic formula or factoring gives us roots of , and
The solution of homogenous equations is written in the form:
so we don't know the constants, but can substitute the values we solved for the roots:
We have three initial values, one for y(t), one for y'(t), and for y''(t) all with t=0
So:
so:
So this can be solved either by substitution or by setting up a 3X3 matrix and reducing. Once you do either of these methods, the values for the constants will be: Then and
This gives a final answer of: