1. **State the problem:** Solve the differential equation $$(D^2 + a^2)y = \sec(ax)$$ using the method of variation of parameters. 2. **Identify the homogeneous equation:** The associated homogeneous equation is $$y'' + a^2 y = 0$$. 3. **Solve the homogeneous equation:** The characteristic equation is $$r^2 + a^2 = 0$$, giving roots $$r = \pm ai$$. The general solution to the homogeneous equation is $$y_h = C_1 \cos(ax) + C_2 \sin(ax)$$. 4. **Set up variation of parameters:** Assume a particular solution of the form $$y_p = u_1(x) \cos(ax) + u_2(x) \sin(ax)$$ where $$u_1$$ and $$u_2$$ are functions to be determined. 5. **Form the system for $$u_1'$$ and $$u_2'$$:** Using the method, we impose $$u_1' \cos(ax) + u_2' \sin(ax) = 0$$ $$-a u_1' \sin(ax) + a u_2' \cos(ax) = \sec(ax)$$ 6. **Solve for $$u_1'$$ and $$u_2'$$:** From the first equation, $$u_1' = -u_2' \frac{\sin(ax)}{\cos(ax)} = -u_2' \tan(ax)$$. Substitute into the second equation: $$-a (-u_2' \tan(ax)) \sin(ax) + a u_2' \cos(ax) = \sec(ax)$$ Simplify: $$a u_2' \tan(ax) \sin(ax) + a u_2' \cos(ax) = \sec(ax)$$ Note $$\tan(ax) = \frac{\sin(ax)}{\cos(ax)}$$, so $$a u_2' \frac{\sin(ax)}{\cos(ax)} \sin(ax) + a u_2' \cos(ax) = \sec(ax)$$ $$a u_2' \frac{\sin^2(ax)}{\cos(ax)} + a u_2' \cos(ax) = \sec(ax)$$ Multiply both terms by $$\cos(ax)$$: $$a u_2' \sin^2(ax) + a u_2' \cos^2(ax) = 1$$ Since $$\sin^2(ax) + \cos^2(ax) = 1$$, $$a u_2' = 1 \implies u_2' = \frac{1}{a}$$. From step 5, $$u_1' = -u_2' \tan(ax) = -\frac{1}{a} \tan(ax)$$. 7. **Integrate to find $$u_1$$ and $$u_2$$:** $$u_2 = \int \frac{1}{a} dx = \frac{x}{a} + C$$ (constant absorbed in homogeneous solution) $$u_1 = -\frac{1}{a} \int \tan(ax) dx = -\frac{1}{a} \cdot \frac{1}{a} \ln|\sec(ax)| + C = -\frac{1}{a^2} \ln|\sec(ax)|$$ 8. **Write the particular solution:** $$y_p = u_1 \cos(ax) + u_2 \sin(ax) = -\frac{1}{a^2} \ln|\sec(ax)| \cos(ax) + \frac{x}{a} \sin(ax)$$ 9. **Final general solution:** $$y = y_h + y_p = C_1 \cos(ax) + C_2 \sin(ax) - \frac{1}{a^2} \ln|\sec(ax)| \cos(ax) + \frac{x}{a} \sin(ax)$$ This completes the solution using variation of parameters.