1. **State the problem:** Solve the differential equation $$y''''' - y'''' = 0$$. 2. **Identify the type of equation:** This is a linear homogeneous differential equation with constant coefficients. 3. **Write the characteristic equation:** Replace derivatives by powers of $r$: $$r^5 - r^4 = 0$$ 4. **Factor the characteristic equation:** $$r^4(r - 1) = 0$$ 5. **Find the roots:** - From $r^4 = 0$, we get a root $r = 0$ with multiplicity 4. - From $r - 1 = 0$, we get a root $r = 1$. 6. **Write the general solution:** For a root $r=0$ with multiplicity 4, the solution terms are: $$C_1 + C_2 x + C_3 x^2 + C_4 x^3$$ For the root $r=1$, the solution term is: $$C_5 e^x$$ 7. **Combine all terms:** $$y = C_1 + C_2 x + C_3 x^2 + C_4 x^3 + C_5 e^x$$ This is the general solution to the differential equation. **Final answer:** $$y = C_1 + C_2 x + C_3 x^2 + C_4 x^3 + C_5 e^x$$