Kansas State University

STEP 1: Plug in $\displaystyle y(x) = \sum_{n=0}^{\infty}a_n x^n $ and compute all the different terms in the equation $$ \begin{align} -3y &= \sum_{ n = 0 }^{ \infty }\,-3 a_{ n } x^{ n }\\ (-x - 5)y' &= \sum_{ n = 1 }^{ \infty }\,-n a_{ n } x^{ n } + \sum_{ n = 1 }^{ \infty }\,-5n a_{ n } x^{ n-1 }\\ (-3x - 1)y'' &= \sum_{ n = 2 }^{ \infty }\,-3n(n-1) a_{ n } x^{ n-1 } + \sum_{ n = 2 }^{ \infty }\,-n(n-1) a_{ n } x^{ n-2 } \end{align} $$ STEP 2: Make the substitutions $ k = n-2 $ and $ j = n-1 $ to make all terms of the form $ x^{\text{index}} $ rather than the $ x^{\text{index}-1} $ or $ x^{\text{index}+2} $ or whatever. $$ \begin{align} -3y&= \sum_{ n = 0 }^{ \infty }\,-3 a_{ n } x^{ n }\\ (-x - 5)y'&= \sum_{ n = 1 }^{ \infty }\,-n a_{ n } x^{ n } + \sum_{ j = 0 }^{ \infty }\,-5(j+1) a_{ j+1 } x^{ j }\\ (-3x - 1)y''&= \sum_{ j = 1 }^{ \infty }\,-3(j+1)j a_{ j+1 } x^{ j } + \sum_{ k = 0 }^{ \infty }\,-(k+2)(k+1) a_{ k+2 } x^{ k }\end{align} $$ STEP 3: Change all the indices to the same letter (I use $ m $) and plug into the equation. $$ \begin{align} (-3x - 1)y'' + (-x - 5)y' - 3y &= \sum_{ m = 1 }^{ \infty }\,-3(m+1)m a_{ m+1 } x^{ m } + \sum_{ m = 0 }^{ \infty }\,-(m+2)(m+1) a_{ m+2 } x^{ m } \\ &+ \sum_{ m = 1 }^{ \infty }\,-m a_{ m } x^{ m } + \sum_{ m = 0 }^{ \infty }\,-5(m+1) a_{ m+1 } x^{ m } \\ &+ \sum_{ m = 0 }^{ \infty }\,-3 a_{ m } x^{ m } \end{align} $$ STEP 4: Collect like terms. (While the general term starts at m=1, it doesn't hurt to separate out the x term as we do below). $$ \begin{align} &(-2a_2 - 5a_1 - 3a_0) + (-6a_{3} - 6a_{2} - 10a_{2} - a_{1} - 3a_{1})x \\ &+ \sum_{ m = 2 }^{ \infty }\,(-(m+2)(m+1)a_{m+2} - 3(m+1)ma_{m+1} - 5(m+1)a_{m+1} - ma_m - 3a_m)x^m = 0 \end{align} $$ STEP 5: Equate coefficients to 0.

Equating the constant term to 0 we get $$ a_2 = \frac{5a_1 + 3a_0}{-2} $$ Equating the linear term to 0 we get $$ a_3 = \frac{16a_2 + 4a_1}{-6} $$ Finally, equating the general term to 0, we find that for $ m \ge 2,$ $$ a_{m+2} = \frac{-(-3(m+1)m - 5(m+1))a_{m+1} - (-m - 3)a_m}{-(m+2)(m+1)} $$ STEP 6: We know that $ a_0 = y(0) = 1 $ and $ a_1 = y'(0) = -5.$ We then plug these values into the formulas found in step 5 to compute the coefficients of the solution.

From the equation for the constant term we get $$ a_2 = \frac{5(-5) + 3(1)}{-2} = 11 $$ From the equation for the linear term we get $$ a_3 = \frac{16(11) + 4(-5)}{-6} = -26 $$ Finally, using the recurrence equation with $ m = 2 $ we get $$ a_4 = \frac{-(-3(2+1)2 - 5(2+1))(-26) - (-2 - 3)(11)}{-(4)(3)} = 803/12 $$

So our solution is $$ y(x) = 1 - 5x + 11x^2 - 26x^3 + (803/12)x^4 + \cdots $$

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