Use the method of undetermined coefficients to get the form of a particular solution to (D +...

Question:

Use the method of undetermined coefficients to get the form of a particular solution to

{eq}(D + 6)^2 (D - 6)^3[y] = 3t^2 e^{-6t}. {/eq}

{eq}y_p {/eq} = A _____ + B _____ + C _____

Method of Undetermined Coefficients:

The method of undetermined coefficients is a method to find the particular solution of a non-homogeneous differential equation.

First, find the complementary solution then find a particular solution.

Complementary solution together with the particular solution makes the general solution.

In complementary solution, we have terms {eq}e^{-6t},te^{-6t},e^{6t},te^{6t},t^2e^{6t} {/eq}.

Particular solution is given as {eq}y_p=A....... + B ....... + C ....... {/eq}

To find blank spaces.

Answer and Explanation:

Given:{eq}(D + 6)^2 (D - 6)^3[y] = 3t^2 e^{-6t} {/eq}

Consider {eq}(D + 6)^2 (D - 6)^3= 0 {/eq}

we get {eq}D=-6,-6,6,6,6 {/eq}

So, complementary solution is {eq}y_c=c_1e^{-6t}+c_2te^{-6t}+c_3e^{6t}+c_4te^{6t}+c_5t^2e^{6t} {/eq}

Also, in RHS of equation {eq}(D + 6)^2 (D - 6)^3[y] = 3t^2 e^{-6t} {/eq}, we have term {eq}t^2 e^{-6t} {/eq} which is also a part of the complementary solution.

So, particular solution is {eq}y_p=Ae^{6t} + B te^{6t} + C t^2e^{6t} {/eq}


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