(Solved): Consider the following differential equation to be solved using a power series. y'' + xy = 0 Using ...

Consider the following differential equation to be solved using a power series.

y'' + xy = 0

Using the substitutiony = 

?	cnxn
n = 0

, find an expression for

c_{k + 2}

in terms of

c_{k ? 1}

for

k = 1, 2, 3  

/0 Points] ZILLDIFFEQ9 6.2.007.EP. Consider the following differential equation to be solved using a power series. \[ y^{\prime \prime}+x y=0 \] Using the substitution \( y=\sum_{n=0}^{\infty} c_{n} x^{n} \), find an expression for \( c_{k+2} \) in terms of \( c_{k-1} \) for \( k=1,2,3 \ldots \) \[ c_{k+2}= \] Find two power series solutions of the given differential equation about the ordinary point \( x=0 \). \[ \begin{array}{l} y_{1}=1+\frac{x^{2}}{2}+\frac{x^{4}}{24}+\ldots \quad \text { and } \quad y_{2}=x+\frac{x^{3}}{6}+\frac{x^{5}}{120}+\ldots \\ y_{1}=1-\frac{x^{3}}{6}+\frac{x^{6}}{180}-\ldots \quad \text { and } y_{2}=x-\frac{x^{4}}{12}+\frac{x^{7}}{504}-\ldots \\ y_{1}=1+x^{2}+\frac{x^{3}}{6}+\ldots \quad \text { and } y_{2}=x+x^{2}+\frac{x^{4}}{12}+\ldots \\ y_{1}=1+\frac{x^{3}}{6}+\frac{x^{6}}{180}-\ldots \quad \text { and } \quad y_{2}=x+\frac{x^{4}}{12}+\frac{x^{7}}{504}-\ldots \\ y_{1}=1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\ldots \quad \text { and } \quad y_{2}=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\ldots \end{array} \]