2. (VP-3) Use variation of parameters to find the general solution to the following equation: y" + y = secx tan x.

Given:

$y\text{'}\text{'}+y=sec x \tan x$

To find:

The solution of the given differential equation using variation of parameters.

Note:

If the differential equation is of the form $ay\text{'}\text{'}+by\text{'}+c=f\left(x\right)$, then $a{m}^2+bm+c=0$ is the corresponding auxiliary equation.

If the root of the auxiliary equation is complex, then the complementary solution is $y_c=e^{ax}\left(A \cos bx + B \sin bx\right)$.

where, the complex roots are $a\pm ib$.

Particular solution:

$y_p=-y_1\int \frac{y_2\left(x\right)f\left(x\right)}{W\left(y_1,y_2\right)}dx+y_2\int \frac{y_1\left(x\right)f\left(x\right)}{W\left(y_1,y_2\right)}dx$.

Where, $W\left(y_1,y_2\right)=\left|\begin{array}{cc}{y}_1& y_2\\ y_1\text{'}& y_2\text{'}\end{array}\right|$.