Q9E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathit{y}\mathbf{=}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{t}\mathbf{-}{\mathit{t}}^{\mathbf{2}}{\mathit{e}}^{\mathit{t}}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$

Step-by-Step Solution

Answer

Yes, the method of undetermined coefficient together with superposition can be applied.

Step 1: Use the method of undetermined coefficients

Given equation,

$3y\text{'}\text{'}+2y\text{'}+8y=t^2+4t-t^2{e}^{t}sint$ 

 

Here, the given differential equation is non-homogeneous.

 

According to the method of undetermined coefficients, 

 

The method of undetermined coefficients applies only to non-homogeneities that are polynomials, exponentials, sines or cosines, or products of these functions. 

 

And the left-hand side consists of the differential equation with constants coefficients.

Step 2: Final conclusion.

The given,

 

R.H.S. of the equation $t^2$, $4t$and $t^2{e}^{t}sint$ is the combination of polynomials, exponentials, sines or cosines or product of these t function. 

 

From the definition of a polynomial, it should have only non-negative integers as powers. 

 

Therefore, it is a product of a polynomial and an exponential function. 

 

So, the method of undetermined coefficients can be applied.