Question

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation.$\mathbf{2}\mathbf{\omega }\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{-}\mathbf{3}\mathbf{\omega }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{4}{\mathbf{xsin}}^2\mathbf{x}\mathbf{+}\mathbf{4}{\mathbf{xcos}}^2\mathbf{x}$

Short answer

Yes, the method of undetermined coefficients can be applied.

Step-by-step solution

Simplification of the given differential equation. 

Given equation,

$2\mathrm{\omega }\text{'}\text{'}\left(\mathrm{x}\right)-3\mathrm{\omega }\left(\mathrm{x}\right)=4{\mathrm{xsin}}^2\mathrm{x}+4{\mathrm{xcos}}^2\mathrm{x}$

Simplify the above equation,

$\begin{array}{l}2\mathrm{\omega }\text{'}\text{'}\left(\mathrm{x}\right)-3\mathrm{\omega }\left(\mathrm{x}\right)=4\mathrm{x}({\sin }^2\mathrm{x}+{\cos }^2\mathrm{x})\\ 2\mathrm{\omega }\text{'}\text{'}\left(\mathrm{x}\right)-3\mathrm{\omega }\left(\mathrm{x}\right)=4\mathrm{x}......\left(1\right)\end{array}$

Now find the roots of the auxiliary equation.

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

$2\mathrm{\omega }\text{'}\text{'}\left(\mathrm{x}\right)-3\mathrm{\omega }\left(\mathrm{x}\right)=0$

The auxiliary equation for the above equation,

role="math" localid="1654859563585" $\begin{array}{c}2{\mathrm{m}}^2-3=0\\ 2{\mathrm{m}}^2=3\\ \mathrm{m}=\pm \frac{3}{2}\end{array}$

The roots of the auxiliary equation are,

role="math" localid="1654859598798" ${\mathrm{m}}_1=\frac{3}{2},{\mathrm{m}}_2=-\frac{3}{2}$

The complementary solution of the given equation is,

${\mathbf{\omega }}_c\left(\mathbf{x}\right)\mathbf{=}{\mathbf{c}}_1{\mathbf{e}}^{\frac{3}{2}\mathbf{x}}\mathbf{+}{\mathbf{c}}_2{\mathbf{e}}^{\mathbf{-}\frac{3}{2}\mathbf{x}}$.

Final conclusion

The R.H.S. of equation is (4x).

Therefore, the particular solution of the equation,

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)=\mathrm{Ax}+\mathrm{b}$

So, the method of undetermined coefficients can be applied.