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Expert-verified 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.${\mathbf{5}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{3}}{\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{2}}{\mathbf{y}}{\mathbf{=}}{{\mathbf{t}}}^{3}{\mathbf{cos}}{\mathbf{4}}{\mathbf{t}}$

Yes, the method of undetermined coefficients can be applied.

See the step by step solution

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

Given equation,

$5\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}+2\mathrm{y}={\mathrm{t}}^{3}\mathrm{cos}4\mathrm{t}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(1\right)$

Write the homogeneous differential equation of the equation (1),

$5\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}+2\mathrm{y}=0$

The auxiliary equation for the above equation,

$5{\mathrm{m}}^{2}-3\mathrm{m}+2=0$

## Step 2: Now find the roots of an auxiliary equation,

Solve the auxiliary equation,

$\begin{array}{c}5{\mathrm{m}}^{2}-3\mathrm{m}+2=0\\ \mathrm{m}=\frac{-\left(-3\right)±\sqrt{9-4\left(5\right)\left(2\right)}}{2\left(5\right)}\\ \mathrm{m}=\frac{3±\sqrt{-31}}{10}\\ \mathrm{m}=\frac{3±\mathrm{i}\sqrt{31}}{10}\end{array}$

The roots of the auxiliary equation are,

${\mathrm{m}}_{1}=\frac{3+\mathrm{i}\sqrt{31}}{10},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}_{2}=\frac{3-\mathrm{i}\sqrt{31}}{10}$

The complementary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}\left(\mathrm{x}\right)={\mathrm{e}}^{\frac{3}{10}\mathrm{x}}\left[{\mathrm{c}}_{1}\mathrm{cosh}\left(\frac{\sqrt{31}}{10}\right)+{\mathrm{c}}_{2}\mathrm{sinh}\left(\frac{\sqrt{31}}{10}\right)\right]$

## Step 3: Final conclusion:

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 R.H.S. of the differential equation has a finite family.

And the given R.H.S. of the equation ${\mathrm{t}}^{3}\mathrm{cos}\left(4\mathrm{t}\right)$has a final family.

The particular solution of equation (1),

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)=\left({\mathrm{At}}^{3}+{\mathrm{Bt}}^{2}+\mathrm{Ct}+\mathrm{D}\right)\mathrm{cos}\left(4\mathrm{t}\right)+\left({\mathrm{Et}}^{3}+{\mathrm{Ft}}^{2}+\mathrm{Gt}+\mathrm{H}\right)\mathrm{sin}\left(4\mathrm{t}\right)$

Therefore, the method of undetermined coefficients can be applied. ### Want to see more solutions like these? 