Find the general solution of the differential equation y’' - 4y' - 45y = 196e^2t. More information in the description section

Question

Find the general solution of the differential equation

y’' - 4y' - 45y = 196e^2t.

NOTE: Express the general solution in the form c₁y₁(t) + c₂y₂(t) + y_p(t), where c_1, c₂ are arbitrary constants and y_p(t) is the particular solution.

The general solution is : y(t)=

Answer 1

Characteristic equation: r² - 4r - 45 = 0; r = 9 or r = - 5; y₁ = e^9t; y₂ = e^-5t.

y_p = Ae^2t; y_p' = 2Ae^2t; y''_p(t) = 4Ae^2t.

4Ae^2t - 8Ae^2t -4 5Ae^2t = 196e^2t; - 49A = 196; A = - 4

y_p = - 4e^2t. General sol.ution: y = C₁e^9t + C₂e^-5t - 4e^2t