Q28E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

use the annihilator method to determinethe form of a particular solution for the given equation. $y\text{'}\text{'}-6y\text{'}+10y=e^{3x}-x$

Step-by-Step Solution

Answer

$y_p\left(x\right)=c_3+c_{4}x+c_5{e}^{3x}$

Step 1: Solve the homogeneous of the given equation

The homogeneous of the given equation is

$(D^2-6D+10)\left[y\right]=0$

The solution of the homogeneous is

  $y_h\left(x\right)=c_1{e}^{3x}cosx+c_2{e}^{3x}sinx$                                 (1)

Now $e^{3x}-x$ is annihilated by $\left(D^3-3D^2\right)$

Then, every solution to the given nonhomogeneous equation also satisfies 

. $\left(D^3-3D^2\right)(D^2-6D+10)\left[y\right]=0$

Then 

  $y\left(x\right)=c_1{e}^{3x}\cos x+c_2{e}^{3x}\sin x+c_3+c_{4}x+c_5{e}^{3x}$                       (2)

is the general solution to this homogeneous equation 

We know $u\left(x\right)=u_h+u_p$

Comparing (1) & (2)

$y_p\left(x\right)=c_3+c_{4}x+c_5{e}^{3x}$

 

Step 2: Identify the differentiation rules needed

Examine the function to determine which differentiation rules apply: power rule, product rule, quotient rule, chain rule, or special function derivatives.

Step 3: Apply the differentiation rules

Differentiate each term of the function systematically, applying the chain rule for composite functions.

Step 4: Simplify the derivative

Combine like terms, factor where appropriate, and write the derivative in its simplest form.

Step 5: State the final answer

Write the final derivative clearly.

Step 6: Conclude with the answer

$y_p\left(x\right)=c_3+c_{4}x+c_5{e}^{3x}$