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Found in: Page 180

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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

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

See the step by step solution

## 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.