# on 13-Dec-2023 (Wed)

#### Flashcard 7604682755340

Question

For a 2nd order linear inhomogeneous ODE

$$\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)$$

where $$f(x)$$ is a periodic function that is given the period $$L_{0}$$.
Given $$p, q, f(x)$$, to find the ODE's particular integral, [Using Fourier series to solve 2nd linear inhomogeneous ODE particular solution]

1. Calculate the Fourier series of the function $$f(x)$$
2. Try ansatz and using the period $$L_{0}$$: $$\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)$$.
3. Yield the derivatives of $$y(x)$$ and plug them into the original ODE.
order linear inhomogeneous ODE $$\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)$$ where $$f(x)$$ is a periodic function that given the period $$L_{0}$$. To find the ODE's particular integral, <span>1. Calculate the Fourier series of the function $$f(x)$$ 2. Try ansatz and using the period $$L_{0}$$: $$\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)$$. 3. Yield the derivatives of $$y(x)$$ and plug them into the original ODE. 4. Comparing coefficients in front of 1, cosine, sine, respectively, write out the equation system about the ODE. 5. Solve the equation system and write out the particular integral. <span>