Q16E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

find a differential operator that annihilates the given function.

$x^2-e^x$

Step-by-Step Solution

Answer

$D^4-D^3$is the differential operator that annihilates the given function.

Step 1: Differentiate the function continuously

Let the function be $f\left(x\right)=x^2-e^x$

Let $g\left(x\right)=x^2$

Then

$$\begin{gathered} g\text{'}\left(x\right)=2x \\ g\text{'}\text{'}\left(x\right) = 2 \\ g\text{'}\text{'}\text{'}\left(x\right)=0 \\ {D}^3\left[g\right]=0 \end{gathered}$$

Let $h\left(x\right)=-e^x$

Then

$$\begin{gathered} h\text{'}\left(x\right)=-e^x \\ h\text{'}\left(x\right)-h\left(x\right)=0 \\ (D-1)\left[h\right]=0 \end{gathered}$$

Hence $D^3(D-1)\left[f\right]=0$

Then  $D^4-D^3$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

$D^4-D^3$is the differential operator that annihilates the given function.