Q20E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

find a differential operator that annihilates the given function

$x^2{e}^x-xsin4x+x^3$

Step-by-Step Solution

Answer

$D^4(D-1{)}^3{\left(D^2+16\right)}^2$is the differential operator that annihilates the given function.

Step 1: Any nonhomogeneous term of the form f ( x ) = x k e α x c o s β x   OR role="math" localid="1663946799201" x k e α x s i n β x satisfiesrole="math" localid="1663946761280" ( D - α ) 2 + β 2 m [ f ] = 0 for K = 0 , 1 , 2 , . . . , m - 1

Let the function be $f\left(x\right)=x^2{e}^x-x\sin 4x+x^3$

Let $g\left(x\right)=x^2{e}^x$

Then

$(D-1{)}^3\left[g\right]=0$

Let $h\left(x\right)=x{e}^{-5x}\sin 3x$

Then

${\left(D^2+4^2\right)}^2\left[h\right]=0$            

Let $i\left(x\right)=x^3$

Then

  $D^4\left[i\right]=0$          

Hence

$$\begin{gathered} (D-1{)}^3{\left(D^2+16\right)}^2{D}^4\left[f\right]=0 \\ \Rightarrow D^4(D-1{)}^3{\left(D^2+16\right)}^2\left[f\right]=0 \end{gathered}$$

Then $D^4(D-1{)}^3{\left(D^2+16\right)}^2$ is the differential operator that annihilates the given function.

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

$D^4(D-1{)}^3{\left(D^2+16\right)}^2$is the differential operator that annihilates the given function.