Science:Math Exam Resources/Courses/MATH103/April 2009/Question 03 (a)

MATH103 April 2009

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Question 03 (a)
Consider the differential equation ${\frac {dx}{dt}}={\frac {x^{2}-2}{2x}}$ 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!

Hint
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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. To solve the differential equation $\displaystyle {\frac {dx}{dt}}={\frac {x^{2}-2}{2x}}$ with separation of variables we rewrite the equation as $\displaystyle {\frac {2xdx}{x^{2}-2}}=dt$ and integrate both sides $\int {\frac {2x}{x^{2}-2}}dx=\int 1dt.$ The integral on the right hand side simply gives t+c. The integrand on the left hand side is of the form ${\frac {f'(x)}{f(x)}}$, with f(x) = x2-2, which has anti-derivative $\ln |f(x)|=\ln |x^{2}-2|$. If you don't see this easily you can simply use the substitution u = x2-2. Either way, we obtain $\ \ln(|x^{2}-2|)=t+c$ for a constant c. To solve this equation for x we first take the exponential on both sides $\displaystyle |x^{2}-2|=e^{t+c}=e^{c}\,e^{t}=de^{t},$ where d is another constant, which has to be positive. Finally, to remove the absolute value we write $\pm (x^{2}-2)=\displaystyle |x^{2}-2|=de^{t},$ i.e. $x^{2}-2=\pm de^{t}={\tilde {c}}e^{t}$ for another constant ${\tilde {c}}$ that has no restriction on its sign (i.e. can be positive or negative). Now we can finally solve for x and obtain $\ x(t)=\pm {\sqrt {{\tilde {c}}e^{t}+2}}.$

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