Questions: Find a particular solution to the differential equation using the Method of Undetermined Coefficients. x''(t)-18x'(t)+81x(t)=te^(9t) What is the auxiliary equation associated with the given differential equation? (Type an equation using r as the variable.) A solution is xp(t)=.

Solution Steps

To find a particular solution to the given differential equation using the Method of Undetermined Coefficients, we first solve the associated homogeneous equation to find the complementary solution. The auxiliary equation is derived from the homogeneous part of the differential equation. Then, we propose a form for the particular solution based on the non-homogeneous term \( t e^{9t} \) and determine the coefficients by substituting back into the original equation.

The given differential equation is

\[ x''(t) - 18x'(t) + 81x(t) = t e^{9t}. \]

To find the auxiliary equation, we consider the homogeneous part:

\[ x''(t) - 18x'(t) + 81x(t) = 0. \]

The auxiliary equation is obtained by substituting \( x(t) = e^{rt} \):

\[ r^2 - 18r + 81 = 0. \]

We can factor the auxiliary equation:

\[ (r - 9)^2 = 0. \]

Thus, the roots are \( r = 9 \) with multiplicity 2. The complementary solution is given by:

\[ x_c(t) = (C_1 + C_2 t)e^{9t}, \]

where \( C_1 \) and \( C_2 \) are constants.

For the non-homogeneous term \( t e^{9t} \), we propose a particular solution of the form:

\[ x_p(t) = t(A + Bt^2)e^{9t}. \]

Substituting \( x_p(t) \) into the original differential equation and solving for the coefficients \( A \) and \( B \) leads to the particular solution:

\[ x_p(t) = t\left(C_2 + \frac{t^2}{6}\right)e^{9t}. \]

Final Answer