Q12E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

find a differential operator that annihilates the given function.

$3x^2-6x+1$

Step-by-Step Solution

Answer

$D^3$ is the differential operator that annihilates the given function.

Step 1: Differentiate the function continuously

Let the function be $f\left(x\right)=3x^2-6x+1$

Then

$$\begin{gathered} f\text{'}\left(x\right)=6x-6 \\ f\text{'}\text{'}\left(x\right)=6 \\ f\text{'}\text{'}\text{'}\left(x\right)=0 \end{gathered}$$

Hence $D^3\left[f\right]=0$

Then $D^3$ is the differential operator that annihilates the given function.

 

 

 

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

$D^3$ is the differential operator that annihilates the given function.