1. **State the problem:** Solve the differential equation $$y'' - 7y' + 10y = -t^2 e^{2t}$$ with initial conditions $$y(0) = 3$$ and $$y'(0) = 9$$. 2. **General solution form:** The solution is given by $$y = y_{IVP} + (h(t) * f(t))$$ where: - $$y_{IVP}$$ is the solution to the homogeneous equation with initial values. - $$h(t)$$ is the impulse response (solution to the homogeneous equation with zero initial conditions and delta forcing). - $$f(t) = -t^2 e^{2t}$$ is the forcing function. 3. **Solve the homogeneous equation:** The associated homogeneous equation is: $$y'' - 7y' + 10y = 0$$ Characteristic equation: $$r^2 - 7r + 10 = 0$$ Factor: $$(r - 5)(r - 2) = 0$$ Roots: $$r = 5, 2$$ General homogeneous solution: $$y_h = C_1 e^{5t} + C_2 e^{2t}$$ 4. **Apply initial conditions to find $$y_{IVP}$$:** Given $$y(0) = 3$$ and $$y'(0) = 9$$: - At $$t=0$$: $$y_h(0) = C_1 + C_2 = 3$$ - Derivative: $$y_h' = 5C_1 e^{5t} + 2C_2 e^{2t}$$ At $$t=0$$: $$y_h'(0) = 5C_1 + 2C_2 = 9$$ Solve the system: $$\begin{cases} C_1 + C_2 = 3 \\ 5C_1 + 2C_2 = 9 \end{cases}$$ Multiply first by 2: $$2C_1 + 2C_2 = 6$$ Subtract from second: $$(5C_1 + 2C_2) - (2C_1 + 2C_2) = 9 - 6$$ $$3C_1 = 3 \Rightarrow C_1 = 1$$ Then: $$C_2 = 3 - 1 = 2$$ So: $$y_{IVP} = e^{5t} + 2 e^{2t}$$ 5. **Find the impulse response $$h(t)$$:** Impulse response solves: $$y'' - 7y' + 10y = \delta(t)$$ with zero initial conditions. The impulse response is the solution to the homogeneous equation with initial conditions: $$h(0) = 0, h'(0) = 1$$ General form: $$h(t) = A e^{5t} + B e^{2t}$$ Apply initial conditions: $$h(0) = A + B = 0$$ $$h'(t) = 5A e^{5t} + 2B e^{2t}$$ At $$t=0$$: $$h'(0) = 5A + 2B = 1$$ From first: $$B = -A$$ Substitute: $$5A + 2(-A) = 1 \Rightarrow 3A = 1 \Rightarrow A = \frac{1}{3}$$ Then: $$B = -\frac{1}{3}$$ So: $$h(t) = \frac{1}{3} e^{5t} - \frac{1}{3} e^{2t}$$ 6. **Forcing function:** $$f(t) = -t^2 e^{2t}$$ 7. **Final solution:** $$y = y_{IVP} + (h * f) = e^{5t} + 2 e^{2t} + \left( \frac{1}{3} e^{5t} - \frac{1}{3} e^{2t} \right) * (-t^2 e^{2t})$$ **Answer:** $$y = e^{5t} + 2 e^{2t} + \left( \frac{1}{3} e^{5t} - \frac{1}{3} e^{2t} \right) * (-t^2 e^{2t})$$