Question

Solve this ordinary differential equation by using the method of variation of parameters y''(x) + y(x) = sec x.

Answer 1

Our first step should be to solve the homogeneous DE associated with this DE:
\(y' ' + y = 0\)
The independent solutions of this equation will be useful in the variatioin of parameters solution method.

Answer 2

The method of finding the particular solution is using this structure.
\(Y_p = u_1 y_1 + u_2 y_2 \) For linearly independent solutions to the homogeneous DE \(y_1\) and \(y_2\) (we find from above)

We could work out the solution with some manipulations, or there are also the formulas for \(u_1\) and \(u_2\).
\(y" + p(x) y' + q(x) y = f(x) \)
\(\to \quad \displaystyle u_1 = - \int \frac{y_2 f(x)}{W(x)} \; \text{d}x \)
\( \quad \quad \ \displaystyle u_2 = \int \frac{y_1 f(x)}{W(x)} \; \text{d}x \)
W(x) the wronskian of y_1 and y_2.

Answer 3

Then the general solution to the main problem is:
\( y = c_1 y_1 + c_2 y_2 + Y_p \)

Just piecing together those smaller things we found before.

Answer 4

Would this all make sense? Is there anything I should go over in more detail? :)

Answer 5

thanks dear....

Answer 6

You're welcome!