1. **State the problem:** Find the general solution of the differential equation $$y'' + 7y = 0$$. 2. **Characteristic equation:** Assume a solution of the form $$y = e^{rt}$$. Substitute into the differential equation: $$r^2 e^{rt} + 7 e^{rt} = 0$$ Divide both sides by $$e^{rt}$$ (which is never zero): $$\cancel{e^{rt}}(r^2 + 7) = 0 \Rightarrow r^2 + 7 = 0$$ 3. **Solve for $$r$$:** $$r^2 = -7$$ $$r = \pm \sqrt{-7} = \pm i\sqrt{7}$$ 4. **General solution in complex exponentials:** Using Euler's formula, the general solution is: $$y(t) = c_1 e^{i\sqrt{7}t} + c_2 e^{-i\sqrt{7}t}$$ 5. **Convert to sines and cosines:** By Euler's formula: $$e^{i\theta} = \cos \theta + i \sin \theta$$ $$e^{-i\theta} = \cos \theta - i \sin \theta$$ So the general solution can be written as: $$y(t) = A \cos(\sqrt{7}t) + B \sin(\sqrt{7}t)$$ where $$A$$ and $$B$$ are constants related to $$c_1$$ and $$c_2$$. **Final answers:** - Complex exponential form: $$y(t) = c_1 e^{i\sqrt{7}t} + c_2 e^{-i\sqrt{7}t}$$ - Sine and cosine form: $$y(t) = A \cos(\sqrt{7}t) + B \sin(\sqrt{7}t)$$