1. **State the problem:** Solve the differential equation $$y^{(5)} - y^{(4)} = 0$$ where $$y^{(5)}$$ is the fifth derivative of $$y$$ and $$y^{(4)}$$ is the fourth derivative. 2. **Rewrite the equation:** The equation can be written as $$y^{(5)} = y^{(4)}$$. 3. **Characteristic equation:** Assume a solution of the form $$y = e^{rx}$$. Then derivatives become $$r^5 e^{rx}$$ and $$r^4 e^{rx}$$. Substitute into the equation: $$r^5 e^{rx} - r^4 e^{rx} = 0$$ Divide both sides by $$e^{rx}$$ (nonzero): $$r^5 - r^4 = 0$$ 4. **Factor the characteristic polynomial:** $$r^4(r - 1) = 0$$ 5. **Find roots:** - $$r^4 = 0$$ gives root $$r=0$$ with multiplicity 4. - $$r - 1 = 0$$ gives root $$r=1$$. 6. **General solution:** For repeated roots, the solution includes terms multiplied by powers of $$x$$: $$y = C_1 + C_2 x + C_3 x^2 + C_4 x^3 + C_5 e^{x}$$ where $$C_1, C_2, C_3, C_4, C_5$$ are constants determined by initial conditions. **Final answer:** $$y = C_1 + C_2 x + C_3 x^2 + C_4 x^3 + C_5 e^{x}$$