Answer:
y = C₂ e²ᵗ + e⁻ᵗ [C₁ - (5t/3)]
Step-by-step explanation:
y" - y' - 2y = 5e⁻ᵗ
The total solution of this differential equation will be a sum of its complimentary and particular solutions.
y = y꜀ + yₚ
The complimentary or homogenous solution is a solution of
y" - y' - 2y = 0
y꜀ = Ay₁ + By₂
Solving this part in the attached image, we obtain
y₁ = e⁻ᵗ and y₂ = e²ᵗ
The particular solution can be obtained using the method of variation of parameters and the method of undetermined coefficients.
The method of variation of parameters is used and explained in the page 1, 2 and page 3 of the attached images.
The method of undetermined coefficients is then used to check if the solution obtained is correct.
With y = C₂ e²ᵗ + e⁻ᵗ [C₁ - (5t/3)],
y' and y'' were obtained and put in the equation, and the answer obtained was 5e⁻ᵗ.
So, Proved!
y = C₂ e²ᵗ + e⁻ᵗ [C₁ - (5t/3)]
Hope this helps!!!
