Q14E - 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.

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

Step-by-Step Solution

Answer

Yes, the method of undetermined coefficients together with superposition can be applied.

 

Step 1: Use the method of undetermined coefficients.

Given equation,

 

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

 

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. 

 

One gets, that the left-hand side consists of the differential equation with constants coefficients. So, there is no problem. 

 

And the right-hand side mathematical expression is a linear combination of algebraic and trigonometric functions. 

Step 2: The method of the Superposition Principle

Let ${\mathrm{y}}_1$ be a solution of the differential equation, $\mathrm{ay}\text{'}\text{'}+\mathrm{by}\text{'}+\mathrm{cy}={\mathrm{f}}_1\left(\mathrm{t}\right)$  and ${\mathrm{y}}_2$ be a solution of the differential equation, $\mathrm{ay}\text{'}\text{'}+\mathrm{by}\text{'}+\mathrm{cy}={\mathrm{f}}_2\left(\mathrm{t}\right)$.

 

Then for any constants ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$, the function k,  ${\mathrm{k}}_1{\mathrm{y}}_1+{\mathrm{k}}_2{\mathrm{y}}_2$ is a solution to the differential equation, 

 $\mathrm{ay}\text{'}\text{'}+\mathrm{by}\text{'}+\mathrm{cy}={\mathrm{k}}_1{\mathrm{f}}_1\left(\mathrm{t}\right)+{\mathrm{k}}_2{\mathrm{f}}_2\left(\mathrm{t}\right)\dots \left(1\right)$

Step 3: Conclusion.

Given equation,

 $2\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-4\mathrm{y}=2\mathrm{t}+{\sin }^2\mathrm{t}+3\dots \left(2\right)$

One knows that,

${\sin }^2\mathrm{t}=\frac{1-\cos \left(2\mathrm{t}\right)}{2}$

Substitute the above formula in the equation (2),

$\begin{array}{c}2\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-4\mathrm{y}=2\mathrm{t}+\frac{1-\cos \left(2\mathrm{t}\right)}{2}+3\\ 2\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-4\mathrm{y}=2\mathrm{t}+\frac{7}{2}-\frac{\cos \left(2\mathrm{t}\right)}{2}....\left(3\right)\end{array}$

Compare equations (1) and (3),

$\begin{array}{c}{\mathrm{y}}_1=2\mathrm{t}+\frac{7}{2}\\ {\mathrm{y}}_2=\frac{\cos \left(2\mathrm{t}\right)}{2}\end{array}$

Therefore, the method of undetermined coefficients together with superposition can be applied.