Q16E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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.$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{t}}^2{\mathbf{e}}^{-t}\mathbf{sint}\mathbf{-}\mathbf{8}\mathbf{tcos}\mathbf{3}\mathbf{t}\mathbf{+}{\mathbf{10}}^t$

Step-by-Step Solution

Answer

Yes, the method of undetermined coefficients together with superposition can be applied to find a particular solution to the given equation.

 

Step 1: Use the method of undetermined coefficients.

Given equation,

 

$2\mathrm{y}\text{'}\text{'}-\mathrm{y}\text{'}+6\mathrm{y}={\mathrm{t}}^2{\mathrm{e}}^{-\mathrm{t}}\mathrm{sint}-8\mathrm{tcos}3\mathrm{t}+{10}^{\mathrm{t}}$

 

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, sines, cosines, or products of these functions. 

 

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

Step 2: Conclusion

The given R.H.S. of the equation ${\mathrm{t}}^2{\mathrm{e}}^{-\mathrm{t}}\mathrm{sint}$, $8\mathrm{tcos}3\mathrm{t}$ and  ${10}^{\mathrm{t}}={\mathrm{e}}^{\mathrm{tlog}\left(10\right)}$ is the combination of polynomials, exponentials, sines, or cosines or product of these functions. 

 

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

 

Therefore, it is a product of polynomial, trigonometric and exponential functions. 

 

So, the method of undetermined coefficients can be applied.