Chapter 4: Q6E (page 180)

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

Short Answer

Expert verified

Yes, the method of undetermined coefficients can be applied.

Step by step solution

Simplification of the given differential equation.

Given equation,

$2 ω'' (x) - 3 ω (x) = 4 {xsin}^{2} x + 4 {xcos}^{2} x$

Simplify the above equation,

$\begin{array}{l} 2 ω'' (x) - 3 ω (x) = 4 x (\sin^{2} x + \cos^{2} x) \\ 2 ω'' (x) - 3 ω (x) = 4 x . ..... (1) \end{array}$

Now find the roots of the auxiliary equation.

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

$2 ω'' (x) - 3 ω (x) = 0$

The auxiliary equation for the above equation,

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

The roots of the auxiliary equation are,

role="math" localid="1654859598798" $m_{1} = \frac{3}{2}, m_{2} = - \frac{3}{2}$

The complementary solution of the given equation is,

$ω_{c} (x) = c_{1} e^{\frac{3}{2} x} + c_{2} e^{- \frac{3}{2} x}$ .

Final conclusion

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

Therefore, the particular solution of the equation,

$y_{p} (x) = Ax + b$

So, the method of undetermined coefficients can be applied.

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

Short Answer

Step by step solution

Simplification of the given differential equation.

Now find the roots of the auxiliary equation.

Final conclusion

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Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation.2ω''(x)-3ω(x)=4xsin2x+4xcos2x

Short Answer

Step by step solution

Simplification of the given differential equation.

Now find the roots of the auxiliary equation.

Final conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Applied Mathematics

Statistics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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