Science:Math Exam Resources/Courses/MATH103/April 2011/Question 06 (b)
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Question 06 (b)
Consider the differential equation
Solve for given the initial value .
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
This differential equation is separable.
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The differential equation is separable; that is,
On the right hand side,
for some constant On the left hand side, we factor and use partial fractions.
This must be true for all . In particular, at , , hence, . At , , hence . Therefore,
for some constant The initial condition is so at least for small , we know that . This means that y-1 is negative and y+1 is positive. Hence
Equating the left and right sides of our first equation, we have
where . From the initial condition, , must be equal to 1. Hence,
Note that this solution does satisfy for all .
Remember: It is important (and usually easy) to check that your solution actually satisfies the differential equation.