Science:Math Exam Resources/Courses/MATH101/April 2007/Question 04 (a)

MATH101 April 2007

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[hide] Question 04 (a)
Full-Solution Problem. Justify your answers and show all your work. Simplification of answers is not required. Find the general solution of the differential equation $4y''+y'=0$ .

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.

[show] Hint 1
Try bringing the $y'$ term over first. Then integrating both sides once is easy.

[show] Hint 2
Next, try an integrating factor.

[show] Hint 3
Multiply both sides by $e^{x/4}$ . Then integrate.

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.

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.
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[show] Solution
Bringing the $y'$ term over we have $\displaystyle 4y''=-y'$ Integrating both sides yields $\displaystyle 4y'=-y+C$ Now, bring the y term to the left and dividing by ${\frac {1}{4}}$ , we have $y'+{\frac {y}{4}}={\frac {C}{4}}$ Now, multiplying both sides by $e^{x/4}$ yields $e^{x/4}y'+e^{x/4}{\frac {y}{4}}=e^{x/4}{\frac {C}{4}}$ The left hand side is now a product rule derivative, namely $(e^{x/4}y)'=e^{x/4}{\frac {C}{4}}$ Integrating both sides yields $e^{x/4}y=\int e^{x/4}{\frac {C}{4}}\,dx=e^{x/4}C+D$ Dividing both sides by $e^{x/4}$ gives $\displaystyle y=C+De^{-x/4}$ completing the question.