Q1E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}{\mathbf{t}}^{-1}{\mathbf{e}}^t$

Step-by-Step Solution

Answer

No, the method of undetermined coefficients can’t be applied.

Step 1: Use the method of undetermined coefficients to find a particular solution to the given differential equation.

Given equation,

 

$\mathrm{y}\text{'}\text{'}+2\mathrm{y}\text{'}-\mathrm{y}={\mathrm{t}}^{-1}{\mathrm{e}}^{\mathrm{t}}$

 

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, sine, or cosine, or products of these functions. 

Step 2: Final conclusion.

The R.H.S. of the equation ${\mathbf{t}}^{-1}{\mathbf{e}}^t$ is not in the form of polynomials, exponentials, sine or cosine, or the product of these t functions. 

 

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

 

And we know that, ${\mathbf{t}}^{\mathbf{-}\mathbf{1}}$ is not a polynomial. 

 

Therefore, it is not a product of a polynomial and an exponential function.

 

So, the method of undetermined coefficients cannot be applied.