Find a particular solution to the differential equation y''−7y'+12y=−12t^2+74t−25?

To get the particular solution, an easy way would be to use the method of undetermined coefficients.

Let Yp (particular solution) = A*(t^2) + B*(t) + C, where A, B, and C are constants to be solved.

Yp' = 2A*t + B

Yp'' = 2A

Now we plug in Yp, Yp', and Yp'' into the differential equation.

2A - 7*(2A*t + B) + 12*[A*(t^2) + B*(t) + C] = -12*(t^2) + 74*t - 25

Simply and rearrange.

12A*(t^2) + (-14A + 12B)*t + (2A - 7B + 12C) = -12*(t^2) + 74*t - 25

Equate the powers of 't'

12A*(t^2) = -12*(t^2) --> A = -1

(-14A + 12B)*t = 74*t --> B = 5

(2A - 7B + 12C) = -25 --> C = 1

So Yp = -(t^2) + 5*t + 1

You were very close, hope that helped.

of direction you will detect a particular answer. You look for it interior the fashion of a polynomial of degree 3 because of the fact that could be a linear ODE with consistent coefficients and your information is a polynomial of degree 3 . so which you have a polynomial of the kind p(t) = At^3 + B t^2 +Ct +D, you compute the derivatives p'(t) and p''(t). Plug into your equation and detect A,B,C,D one after the different in this order. You get A = -12 and that i help you end. you prefer under 5 mns. Oh i did no longer comprehend that this exchange into hidden someplace on the tip of the previous solutions... ok