QQuestionMathematics
QuestionMathematics
Find a linear differential operator that annihilates the given function (use D for the differential operator)
e^-x + 7xe^x - x^2 e^x
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Answer
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Step 1:: Identify the given function
The given function is $f(x) = e^{-x} + 7xe^{x} - x^2 e^{x}$.
Step 2:: Find the derivative of the function
\frac{df(x)}{dx} = -e^{-x} + 7e^{x} + 7xe^{x} - 2x e^{x}
In order to find a linear differential operator that annihilates the given function, we first need to find the derivative of the function.
Step 3:: Factor out $e^{x}$
\frac{df(x)}{dx} = e^{x} (-1 + 7x - 2x^2 + 7 - x)
Step 4:: Set the derivative equal to zero and solve for $x$
x = 1, 3
To find the annihilator, we need to find a linear differential operator that makes the derivative equal to zero. Rearranging the terms, we have: Dividing the entire equation by 2, we get: Factoring the quadratic equation, we get:
Step 5:: Form the annihilator
(D - 1)(D - 3)
A linear differential operator that annihilates the given function is:
Final Answer
The annihilator for the given function $f(2$) = e^{-x} + 7xe^{x} - x^2 e^{x}$ is $(D - 1)(D - 3)$.
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