Final answer:
To solve the differential equation y'' - y = sec(x) tan(x) by variation of parameters, first find the complementary solution of the homogeneous equation y'' - y = 0. Then, use variation of parameters to find the particular solution.
Step-by-step explanation:
To solve the given differential equation by variation of parameters, we first find the complementary solution of the homogeneous equation y'' - y = 0, which is y_c(x) = c1e^x + c2e^{-x}.
Next, we find the particular solution y_p(x) using variation of parameters. We assume y_p(x) = u1(x)e^x + u2(x)e^{-x} and solve for u1(x) and u2(x) using the formulas:
- u1(x) = -\int \frac{y_c(x)R_2(x)}{W(y_1,y_2)} dx
- u2(x) = \int \frac{y_c(x)R_1(x)}{W(y_1,y_2)} dx
where y_1(x) and y_2(x) are the solutions of the homogeneous equation, and R_1(x) and R_2(x) are the right-hand side functions.