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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.${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{6}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{4}}{\mathbf{y}}{\mathbf{=}}{\mathbf{4}}{\mathbf{sin}}{\mathbf{3}}{\mathbf{t}}{\mathbf{-}}{{\mathbf{t}}}^{2}{{\mathbf{e}}}^{3t}{\mathbf{+}}\frac{1}{t}$

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,

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{-}{\mathbf{t}}^{2}{\mathbf{e}}^{3t}\mathbf{+}\frac{1}{t}$

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, cosine, or products of these functions.

## Step 2: Conclusion

The given R.H.S. of the equation $\frac{1}{\mathrm{t}}$ is not the combination of polynomials t.

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

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

So, the method of undetermined coefficients cannot be applied.