Science:Math Exam Resources/Courses/MATH101/April 2008/Question 04 (a)
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Question 04 (a) 

Solve the initialvalue problem 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

This is a homogeneous second order differential equation. 
Hint 2 

Find the roots of the auxiliary polynomial, namely the roots of

Hint 3 

Then solve for the constants using the initial conditions. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Setting up the auxiliary polynomial and computing the roots gives
and so the roots are and . Thus, the general solution to this differential equation is Now, we plug in the initial conditions. Finding the derivative, we have
At the initial condition , we have
Clearing denominators and simplifying yields
At the initial condition , we have
We solve for the constants. Taking times the second equation and adding to the first yields
and so . Taking times the second equation and adding it to the first yields
and so . Thus, the solution is completing the question. 