### Select your language

Suggested languages for you:

Americas

Europe

Q12E

Expert-verified
Found in: Page 186

### 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 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. ${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{ty}}{\mathbf{=}}{{\mathbf{e}}}^{t}{\mathbf{+}}{\mathbf{7}}$

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

See the step by step solution

## Step 1: Use the method of undetermined coefficients.

Given equation,

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{=}{\mathbf{e}}^{t}\mathbf{+}\mathbf{7}$

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, cosine, or products of these functions.

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

## Step 2: Conclusion.

The L.H.S. of the equation has a non-constant coefficient, $\mathrm{t}$.

Thus, the method of undetermined coefficients cannot be applied.