Differential Equations

Free Version

Upgrade subject to access all content

Moderate

First Order Linear Equation and Superposition Principles

DIFFEQ-YQH4KG

Consider the first order linear equation:

$$y'+2y=x^2$$

According to the superposition principles, we can first find a particular solution $y_p$ of the given equation and then solve the homogeneous equation:

$$y'+2y=0$$

The homogeneous equation can be solved , for example, using the characteristic equation $r+2=0$ to find $y_h=Ce^{-2x}$. (Of course, you can also use the technique for separable equations to find $y_h$). The original equation can then be solved.

Select ALL the correct statements regarding this method.

A

One particular solution is obtained by the constant solution, or $y'=0$. Hence, $2y_p=x^2$ or $y_p=\cfrac{1}{2}x^2$.

B

One particular solution can be obtained by trying quadratic function $y_p=ax^2+bx+c$, yielding $y_p=\frac{1}{2}x^2-\cfrac{1}{2}x+\frac{1}{4}$.

C

The full list of solutions to $y'+2y=x^2$ is $y=Ce^{-2x}$.

D

The full list of solutions to $y'+2y=x^2$ is $y=Ce^{-2x}+\cfrac{1}{2}x^2$.

E

The full list of solutions to $y'+2y=x^2$ is $y=Ce^{-2x}+\cfrac{1}{2}x^2-\cfrac{1}{2}x+\cfrac{1}{4}$.