Questions: Determine the form of a particular solution for the differential equation. Do not solve. y'' - 12y' + 37y = e^(7t) + t sin(3t) - cos(3t) The form of a particular solution is yp(t) =

Determine the form of a particular solution for the differential equation. Do not solve.

y'' - 12y' + 37y = e^(7t) + t sin(3t) - cos(3t)

The form of a particular solution is yp(t) =

Transcript text: Determine the form of a particular solution for the differential equation. Do not solve. \[ y^{\prime \prime}-12 y^{\prime}+37 y=e^{7 t}+t \sin 3 t-\cos 3 t \] The form of a particular solution is $y_{p}(t)=$ $\square$ (Do not use d, D, e, $E_{;}, i$, or $I$ as arbitrary constants since these letters already have defined meanings.)

Solution

Solution Steps

To determine the form of a particular solution for the given non-homogeneous linear differential equation, we need to consider the right-hand side of the equation, which consists of three terms: $e^{7t}$, $t \sin 3t$, and $-\cos 3t$. For each term, we propose a form for the particular solution. For $e^{7t}$, we use $Ae^{7t}$. For $t \sin 3t$, we use $(Bt + C)\sin 3t + (Dt + E)\cos 3t$. For $-\cos 3t$, we use $F\cos 3t + G\sin 3t$. We combine these forms to propose the complete form of the particular solution.

Step 1: Identify the Non-Homogeneous Terms

The given differential equation is: \[ y'' - 12y' + 37y = e^{7t} + t \sin 3t - \cos 3t \] The non-homogeneous terms on the right-hand side are $e^{7t}$, $t \sin 3t$, and $-\cos 3t$.

Step 2: Propose the Form for Each Non-Homogeneous Term

For $e^{7t}$, propose a particular solution of the form $Ae^{7t}$.
For $t \sin 3t$, propose a particular solution of the form $(Bt + C)\sin 3t + (Dt + E)\cos 3t$.
For $-\cos 3t$, propose a particular solution of the form $F\cos 3t + G\sin 3t$.

Step 3: Combine the Proposed Forms

Combine the proposed forms for each non-homogeneous term to form the complete particular solution: \[ y_p(t) = A e^{7t} + (B t + C) \sin 3t + (D t + E) \cos 3t + F \cos 3t + G \sin 3t \]

Step 4: Simplify the Particular Solution

Combine like terms in the trigonometric components: \[ y_p(t) = A e^{7t} + (B t + C + G) \sin 3t + (D t + E + F) \cos 3t \]