Science:Math Exam Resources/Courses/MATH101/April 2014/Question 06
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Question 06 

Long Problem. Show your work. No credit will be given for the answer without the correct accompanying work. Find the solution of the differential equation that satisfies . 
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 

How can we solve a differential equation? Think about the separation of variables. 
Hint 2 

Use partial fraction when you compute the integral. 
Hint 3 

Keep track of the absolute value and determine the solution which satisfy the initial data. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. By the separation of variables,
For the integral, we can decompose and therefore, we have,
This implies The initial condition is , and so we get . Thus, Furthermore, when we use the initial data both quantities are positive and so we can drop the absolute value signs: 