Science:Math Exam Resources/Courses/MATH101/April 2017/Question 01 (d)

MATH101 April 2017

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Question 01 (d)
Let $y$ be the solution to the differential equation ${\frac {dy}{dx}}={\frac {\cos x}{2y}}$ satisfying $y(0)={\sqrt {3}}.$ What is $y\left({\frac {\pi }{2}}\right)$? Simplify your answer completely.

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
Use chain rule to integrate on one side with respect to $y$ and the other with respect to $x$.

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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. Since the differential equation ${\frac {dy}{dx}}={\frac {\cos x}{2y}}$ is separable, we can rewrite it as $ydy={\frac {\cos xdx}{2}}$. We integrate both sides $\int y{dy}=\int {\frac {\cos xdx}{2}}$ and thus ${\frac {y^{2}}{2}}={\frac {\sin x}{2}}+C$. This yields $y(x)=\pm {\sqrt {\sin x+2C}}$. Since $y(0)={\sqrt {3}}>0$ , this means that we must take the positive root and that $2C=3$. Evaluate at $x={\frac {\pi }{2}}$ and obtain Answer: $\color {blue}y\left({\frac {\pi }{2}}\right)=2$