Science:Math Exam Resources/Courses/MATH103/April 2014/Question 07 (c)

MATH103 April 2014

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Question 07 (c)
Suppose the function $y(x)=\sum _{n=0}^{\infty }c_{n}x^{n}$ given by a power series solves following differential equation ${\frac {dy}{dx}}=2+x^{2}y$ with the initial value $y(0)=1$. Determine $c_{3}$ and $c_{4}$. Hint: Plug the series into the differential equation. Work must be shown for full marks. Simplify fully.

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.

Hint 1
After plugging the series into the differential equation, differentiate the left-hand side term-by-term.

Hint 2
Following the first hint yields an equation involving the equality of two series. Compare the coefficients of the monomials $x^{0},x^{1},x^{2},\ldots$ in the two series.

Hint 3
Don’t forget to use the given initial condition.

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. 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. Substituting $y(x)=\sum _{n=0}^{\infty }c_{n}x^{n}$ into ${\frac {dy}{dx}}=2+x^{2}y$ yields $\sum _{n=1}^{\infty }nc_{n}x^{n-1}=2+\sum _{n=0}^{\infty }c_{n}x^{n+2}.$ Writing out the first few terms explicitly on both sides, we have $c_{1}+2c_{2}x+3c_{3}x^{2}+4c_{4}x^{3}+\cdots =2+c_{0}x^{2}+c_{1}x^{3}+\cdots .$ Comparing the coefficients of $x^{3}$ on both sides, we see that $4c_{4}=c_{1}$. But comparing the coefficients of $x^{0}(=1)$ on both sides, we see that $c_{1}=2$. Thus, $4c_{4}=2$, i. e. $c_{4}={\frac {1}{2}}$. Now comparing the coefficients of $x^{2}$ on both sides, we get $3c_{3}=c_{0}$. Since $c_{0}$ does not appear on the left-hand side, we determine it instead by using the given initial condition $y(0)=1$. That is, setting $x=0$ in $y(x)=\sum _{n=0}^{\infty }c_{n}x^{n}=c_{0}+c_{1}x+\cdots$, we get $y(0)=c_{0}$. Thus, $y(0)=1$ implies $c_{0}=1$. Putting this all together, we get $3c_{3}=c_{0}=1$, i. e. $c_{3}={\frac {1}{3}}$.

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