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Power Series Solutions of Differential Equations

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Solve $y' - xy = 0$ using a power series centered at $x=0$ .

Radius of convergence: $R = \infty$ . Power series solution: $y = Ce^{\frac{x^2}{2}}$

Solve the Airy equation $y'' - xy = 0$ using a power series expansion about $x=0$ .

Radius of convergence: $R = \infty$ . Power series solution: $y = C_1 \text{Ai}(x) + C_2 \text{Bi}(x)$ , where Ai( $x$ ) and Bi( $x$ ) are Airy functions.

Solve $y'' + y = 0$ using a power series centered at $x=0$ .

Radius of convergence: $R = \infty$ . Power series solution: $y = A\cos(x) + B\sin(x)$ .

Solve $y'' + 2xy' + (1-x^2)y = 0$ using a power series about $x=0$ .

Radius of convergence: $R = 1$ . Power series solution: $y = Ce^x$ where $C$ is a constant.

Solve $(1-x)y'' - xy' + y = 0$ for $y(x)$ around $x=0$ using a power series.

Radius of convergence: $R = 1$ . Power series solution: $y = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$ .

Find the power series solution of $y'' - 2xy' + 2ny = 0$ centered at $x=0$ , where $n$ is a nonnegative integer.

Radius of convergence: $R = \infty$ . Power series solution: $y = H_n(x)e^{-x^2}$ , where $H_n(x)$ is the Hermite polynomial of degree $n$ .

Solve $xy'' + 2y' - xy = 0$ using a power series centered at $x=0$ .

Radius of convergence: $R = \infty$ . Power series solution: $y = C_1e^x + C_2xe^x$ where $C_1$ and $C_2$ are constants.

Find the power series solution of $y'' + xy' + y = 0$ around $x=0$ .

Radius of convergence: $R = \infty$ . Power series solution: $y = Ce^{-\frac{x^2}{2}}$ where $C$ is a constant.

Determine the series solution for $y''' - y = 0$ centered at $x=0$ .

Radius of convergence: $R = \infty$ . Power series solution: $y = C_1e^x + C_2e^{-x/2}\cos\left(\frac{\sqrt{3}x}{2}\right) + C_3e^{-x/2}\sin\left(\frac{\sqrt{3}x}{2}\right)$

Determine the series solution for the Legendre differential equation $(1 - x^2)y'' - 2xy' + n(n+1)y = 0$ around $x=0$ .

Radius of convergence: $R = 1$ . Power series solution: $y = P_n(x)$ , where $P_n(x)$ is the Legendre polynomial of degree $n$ .

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