Answer :

Final answer:

To find the particular solution (Yp) of the given ordinary differential equation y'' + y' - 12y = e^2t, use the formula Yp = -t * f * Yr. The complementary solution can be found by solving the associated homogeneous equation. The general solution of the sum of the particular solution and the complementary solution.

Explanation:

The given ordinary differential equation is y'' + y' - 12y = e^2t.

To find the particular solution (Yp), we use the formula Yp = -t * f * Yr, where f is the frequency of the coefficient of the exponential term and Yr is the complementary solution.

In this case, the exponential term has a coefficient of e^2t, so the frequency is 2. The complementary solution can be found by solving the associated homogeneous equation y'' + y' - 12y = 0, which factors into (y-3)(y+4) = 0.

Therefore, the particular solution is Yp = -(t^2)e^2t * (A*cos(2t) + B*sin(2t)), where A and B are constants determined by the initial conditions. The general solution are given differential equation is sum of the particular solution and complementary solution.

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