In my book,

$$y''+p(x)y'+q(x)y=r(x) \tag{1}$$ $$ y''+p(x)y'+q(x)y=0 \tag{2} $$

Here $(1)$ is full equation but $(2)$ is reduced equation. Suppose that in some way we know that $y(x,c_1 ,c_ 2 )$ is the general solution of $(2)$-we expect it to contain two arbitrary constants since the equation is of the second order-and that $y_p (x)$ is a fixed particular solution of $(1)$. If $y(x)$ is any solution whatever of $(1)$, then an easy calculation shows that $y(x) - y_p(x)$ is a solution of $(2)$ : $$(y - y_p )'' + P(x)(y - y_p ) ' + Q(x)(y - y_p )= r(x)-r(x)=0$$

So $y(x)=y(x,c_1,c_2)+y_p$

Here I do not understand what it is meant by $y(x,c_1,c_2)$ and $y_p$,because I don't know what a particular solution means. Can you help me to understand?


The notation $y(x, c_1, c_2)$ is just function notation. It says that $y$ depends on $x, c_1,$ and $c_2$. Without ICs we can't determine the constants.

Now, $y_p$ is called the particular solution. This is used to generate whatever the forcing function on the right side is. This is obtained by variation of parameters, undetermined coefficients, and the like. We obtain $y_p$ in whole by taking $c_1, c_2 = 0$ once we have found our full solution $y = y_c + y_p$, where $y_c$ satisfies $y'' +Py'+Qy = 0$.


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