Differential Equation Power Series Solution Near An Ordinary Point Problems 2

Consider the equation $$\left(4-4 x^{2}\right) y^{\prime \prime}-8 x y^{\prime}+3 y=0$$
Let $$y 1(x)$$ be the solution of the equation $$(1)$$ such that $$y_{1}(0)=8, y_{1}^{\prime}(0)=0$$ . Find the first three nonzero terms of the power series about 0 of $$y_{1}(x) .$$

Let $$y 2(x)$$ be the solution of the equation $$(1)$$ such that $$y 2(0)=0, y_{2}^\prime(0)=1$$ . Use a Wronskian to show that $$\left\{y_{1}, y 2\right\}$$ is a fundamental set for the solutions of the equation $$(1)$$

$$4 y^{\prime \prime}-4 x^{2} y^{\prime \prime}-8 x y^{\prime}+3 y=0$$

$$y=\sum_{n=0}^{\infty} a n x^{n}$$
$$y^{\prime}=\sum_{n=0}^{\infty} a_{n}(n) x^{n-1}$$
$$y^{\prime \prime}=\sum_{n=0}^{\infty} a_{n}(n)(n-1)^{n-2}$$

$$4\left(\sum_{n=0}^{\infty} a_{n}(n)(n-1) x^{n-2}\right)-4 x^{2}\left(\sum_{n=0}^{\infty} a_{n}(n)(n-1) x^{n-2}\right)-8 x \sum_{n=0}^{\infty} a_{n}(n) x^{n-1}+3 \sum_{n=0}^{\infty} a_{n} x^{n}=0$$

$$4 \sum_{n=0}^{\infty} a_{n+2}(n+2)(n+1) x^{n}+\sum_{n=0}^{\infty} a_{n}[-4(n)(n-1)-8 n+3] x^{n}=0$$

$$\sum_{n=0}^{\infty}\left[4\left(a_{n+2}(n+2)(n+1)+a_{n}((-4 n)(n-1)-8 n+3)] x^{n}=0\right.\right.$$

$$4 a_{n+2}(n+2)(n+1)+a_{n}({-4 n})(n-1)-8 n+3=0$$

$$a_{n+2}=\frac{-a_{n}(-4 n^2-4 n+3)}{4(n+2)(n+1)}$$

$$=\frac{a n^{2}+4 n-3}{4(n+2)(n+1)}$$ an $$\quad n \geq 0$$

$$n=0 \longrightarrow a_{2}=\frac{-3 a_{0}}{4(2)(1)}=\frac{-3}{8} a_{0}=-3$$

$$n=1 \rightarrow a_{3}=0=a_{1}$$

$$n=2 \rightarrow a_4=\frac{-21}{16}$$

$$y=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+\cdots$$

$$y=8+0-3 x^{2}+0-\frac{21}{16} x^{4}+\cdots$$

$$y=8-3 x^{2}-\frac{21}{16} x^{4}+\cdots$$

(b) $$W\left(y_{1}, y_{2}\right)(0)=\left|\begin{array}{ll}{y_{1}(0)} & {y_{2}(0)} \\ {y_{1}^{\prime}(0)} & {y_{2}^{\prime}(0)}\end{array}\right|=\left|\begin{array}{ll}{8} & {0} \\ {0} & {1}\end{array}\right|=8-0=8\neq 0$$

\$$\left\{y_{1}, y_{2}\right\}$$ is a fundametal set