Question

Find the two linearly independent solution of d^2y/dx^2 + x dy/dx + y =0 when the series expansion is about the origin ..... Please I will give a medal

Answer 1

$$

\Large y(x) =a_0e^{\frac{-x^2}{2}} +a_1\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}(2n+1)!}{2^n n!}


$$

Answer 2

ill get it started, some of the steps

Answer 3

$$
\text{we want to solve } \\

\Large y ' ' + x y' + y = 0 \\
\text {Assume your solution is } \\
\Large y(x)=\sum_{n=0}^{\infty}a_n x^n\\
\text {then it follows that} \\
\Large y'(x) = \sum_{n=1}^{\infty}n \cdot a_n x^{n-1} \\
\Large y''(x) = \sum_{n=2}^{\infty}n(n-1) \cdot a_n x^{n-2}\\

\text {Substitute: }\Large y ' ' + x y' + y = 0 \\
\large \sum_{n=2}^{\infty}n(n-1) \cdot a_n x^{n-2} + x\sum_{n=1}^{\infty}n \cdot a_n x^{n-1} + \sum_{n=0}^{\infty}a_n x^n=0\\
\large \iff \\
\large \sum_{n=2}^{\infty}n(n-1) \cdot a_n x^{n-2} + \sum_{n=1}^{\infty}n \cdot a_n x^{n} + \sum_{n=0}^{\infty}a_n x^n=0\\
\large \iff \\
\large \sum_{n=0}^{\infty}(n+2)(n+1) \cdot a_{n+2} x^{n} + \sum_{n=1}^{\infty}n \cdot a_n x^{n} + \sum_{n=0}^{\infty}a_{n+2} x^n=0\\
\large \iff\\

\large \sum_{n=0}^{\infty}(n+2)(n+1) \cdot a_{n+2} x^{n} + \sum_{n=1}^{\infty}n \cdot a_n x^{n} + \sum_{n=0}^{\infty}a_n x^n=0\\

$$

Answer 4

thanks so much i understood perfectly. but please can you help me find the initial value problem of that equation when y(0)=1 and dy/dx(0)=0
god bless