Use the method of variation of parameters to find a particular solution of the given differential equation. Then check your answer by using the method of undetermined coefficients. y'' − y' − 2y = 5e−t

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!!!

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step-by-step explanation: