Q24E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

use the annihilator method to determinethe form of a particular solution for the given equation. $\mathit{\theta }\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathit{\theta }\mathbf{=}\mathit{x}{\mathit{e}}^{\mathbf{x}}$

Step-by-Step Solution

Answer

${\theta }_p\left(x\right)=c_{3}x{e}^x+c_4{x}^2{e}^x$

Step 1: Solve the homogeneous of the given equation

The homogeneous of the given equation is

$\left(D^2-1\right)\left[\theta \right]=(D-1)(D+1)\left[\theta \right]=0$

The solution of the homogeneous is

       ${\theta }_h\left(x\right)=c_1{e}^{-x}+c_2{e}^x$                 (1)

Now $x{e}^x$ is annihilated by $(D-1{)}^2$

Then, every solution to the given nonhomogeneous equation also satisfies 

. $(D-1{)}^2(D-1)(D+1)\left[\theta \right]=(D-1{)}^3(D+1)\left[\theta \right]=0$

Then 

 $\theta \left(x\right)=c_1{e}^{-x}+c_2{e}^x+c_{3}x{e}^x+c_4{x}^2{e}^x$    (2)

is the general solution to this homogeneous equation 

We know $u\left(x\right)=u_h+u_p$

Comparing (1) & (2)

${\theta }_p\left(x\right)=c_{3}x{e}^x+c_4{x}^2{e}^x$

 

 

 

 

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

${\theta }_p\left(x\right)=c_{3}x{e}^x+c_4{x}^2{e}^x$