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Chapter 6: Q10E (page 326)
Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
on
The function is linearly independent on .
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. find a differential operator that annihilates the given function.
Let y1x2= Cerx, where C (鈮�0) and r are real numbers,be a solution to a differential equation. Supposewe cannot determine r exactly but can only approximateit by . Let (x) =Cerxand consider the error
(a) If r andare positive, r 鈮犅� , show that the errorgrows exponentially large as x approaches + 鈭�.
(b) If r andare negative, r鈮� , show that the errorgoes to zero exponentially as x approaches + 鈭�.
In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
Constructing Differential Equations. Given three functions that are each three times differentiable and whose Wronskian is never zero on (a, b), show that the equation
is a third-order linear differential equation for which is a fundamental solution set. What is the coefficient of y鈥� in this equation?
Given thatis a fundamental solution set for the homogeneous equation corresponding to the equation
determine a formula involving integrals for a particular solution.
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