(Solved): Series solution of variable-coefficient ODE Consider the variable coefficient ...

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Series solution of variable-coefficient ODE Consider the variable coefficient linear second order homogeneous ODE $y^{??}(x)?x{y}^{?}(x)?y(x)=0.$ 1. The point $x=0$ is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. $y={?}_{m=0}^{?}{a}_m{x}^m$ Write down the first and second derivatives of the power series. 2. Substitute the power series (and its derivatives) into equation (1). Express your answer in the form ${?}_{m=0}^{?}{b}_m{x}^{m?2}+{?}_{m=0}^{?}{c}_m{x}^m=0$ where $b_m$ and $c_m$ are to be written in terms of $m$ and $a_m$. 3. Shift the index on one of the series you found in 2 so that the exponents of $x$ are equal to $m$ in both series. 4. Find a recurrence relation for the coefficients $a_{m+2}$ in terms of $a_m$ and $m$. 5. Use the recurrence relation to find expressions for the coefficients $a_2,a_3,a_4$ and $a_5$ in terms of $a_0$ and $a_1$. 6. Write down the general solution to (1) in the form $y=a_{0}f(x)+a_{1}g(x)$. 7. Determine the values of $a_0$ and $a_1$ for the particular solution satisfying the initial conditions $y(0)=1$ and $y^{?}(0)=0$. 8. Deduce that the solution in 7 can be expressed as an elementary function. Hint: $e^u={?}_{n=0}^{?}\frac{u^n}{n!}.$