Science:Math Exam Resources/Courses/MATH104/December 2012/Question 01 (f)

MATH104 December 2012

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Question 01 (f)
Let $\displaystyle {}y(x)=e^{rx}$ For what value of r does y(x) satisfy $\displaystyle {}y''(x)-4y'(x)+4y(x)=0?$

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 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
Evaluate the first and second derivatives of $y(x)$ and substitute them into the given equation.

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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. Evaluate the first and second derivatives of $y(x)$: $\displaystyle y'(x)=re^{rx},\quad y''(x)=r^{2}e^{rx}.$ Now substitute them into the given equation: ${\begin{aligned}y''(x)-4y'(x)+4y(x)&=0\\r^{2}e^{rx}-4re^{rx}+4e^{rx}&=0\\(r^{2}-4r+4)e^{rx}&=0\end{aligned}}$ Thus, either $e^{rx}=0$ for any $x$ which does not happen since the exponential function never evaluates to 0, or $r^{2}-4r+4=0$. Factoring the quadratic gives $\displaystyle r^{2}-4r+4=(r-2)^{2}=0.$ Thus, the value of $r$ that satisfies the given equation is $\displaystyle {\color {blue}r=2}.$

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