Question

If $y_{p_1}=x$is a particular solution of$y\text{'}\text{'}+y=x$and$y_{p_2}=x^2-2$is a particular solution of$y\text{'}\text{'}+y=x^2$, then a particular solution of$y\text{'}\text{'}+y=x^2+x$is______

Short answer

The particular solution of an DE is$y=x^2+x-2$ .

Step-by-step solution

Define a particular solution of the differential equation.

A solution of the form $y=f\left(x\right)$is a differential equation solution that does not contain any arbitrary constants. The differential equation's general solution is of the form$y=f\left(x\right)$ or $y=ax+b$, with and being arbitrary constants.

A solution of the differential equation is y=-5e-x+10ex.

Let the particular solutions of $y\text{'}\text{'}+y=x$be $y_{p_1}=x$, and of $y\text{'}\text{'}+y=x^2$be$y_{p_2}=x^2-2$ .

Then by using the theorem 4.1.7, the particular solution of an DE $y\text{'}\text{'}+y=\underset{g_2\left(x\right)}{\underbrace{x^2}}+\underset{g_1\left(x\right)}{\underbrace{x}}$is given by the superposition of$y_{p_2}$and$y_{p_1}$.

That is,

$$\begin{gathered} y=y_{p_1}+y_{p_2} \\ =x+x^2-2 \\ =x^2+x-2 \end{gathered}$$