MATH103 April 2009
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Question 03 (a)
Consider the differential equation
Solve the differential equation by separation of variables.
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!
Both integrals you get after separating variables have an elementary anti-derivative. If you don't see it, try a substitution.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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To solve the differential equation
with separation of variables we rewrite the equation as
and integrate both sides
The integral on the right hand side simply gives t+c. The integrand on the left hand side is of the form , with f(x) = x2-2, which has anti-derivative . If you don't see this easily you can simply use the substitution u = x2-2. Either way, we obtain
for a constant c.
To solve this equation for x we first take the exponential on both sides
where d is another constant, which has to be positive. Finally, to remove the absolute value we write
for another constant that has no restriction on its sign (i.e. can be positive or negative). Now we can finally solve for x and obtain
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