Constructing Differential Equations. Given three Functions

Chapter 6, Problem 34E

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QUESTION:

Constructing Differential Equations. Given three functions \(f_{1}(x), f_{2}(x), f_{3}(x)\) that are each three times differentiable and whose Wronskian is never zero on , show that the equation

\(\left.\mid f_{1}(x) f_{2}(x) f_{3}(x) y f_{1}^{\prime}(x) f_{2}(x) f_{3}^{\prime}(x) y f_{1}^{\prime}(x) f_{2}^{\prime \prime}(x) f_{3}^{\prime \prime}(x) y^{\prime \prime} f_{1}^{\prime \prime}(x) f_{2}^{\prime \prime}(x) f_{3}^{\prime \prime}(x) y^{\prime \prime}\right]=0\)

is a third-order linear differential equation for which \(\left(f_{1}, f_{2}, f_{3}\right)\) is a fundamental solution set. What is the coefficient of \(y^{\prime \prime}) in this equation?

Equation transcription:

Text transcription:

f{1}(x), f{2}(x), f{3}(x)

 f{1}(x) f{2}(x) f{3}(x) y f{1}^{prime}(x) f{2}(x) f{3}^{prime}(x) y f{1}^{prime}(x) f_{2}^{prime prime}(x) f{3}^{prime prime}(x) y^{prime prime} f{1}^{prime prime}(x) f{2}^{prime prime}(x) f{3}^{prime prime}(x) y^{prime prime}]=0

(f{1}, f{2}, f{3})

y^{prime prime}

Questions & Answers

QUESTION:

Constructing Differential Equations. Given three functions \(f_{1}(x), f_{2}(x), f_{3}(x)\) that are each three times differentiable and whose Wronskian is never zero on , show that the equation

\(\left.\mid f_{1}(x) f_{2}(x) f_{3}(x) y f_{1}^{\prime}(x) f_{2}(x) f_{3}^{\prime}(x) y f_{1}^{\prime}(x) f_{2}^{\prime \prime}(x) f_{3}^{\prime \prime}(x) y^{\prime \prime} f_{1}^{\prime \prime}(x) f_{2}^{\prime \prime}(x) f_{3}^{\prime \prime}(x) y^{\prime \prime}\right]=0\)

is a third-order linear differential equation for which \(\left(f_{1}, f_{2}, f_{3}\right)\) is a fundamental solution set. What is the coefficient of \(y^{\prime \prime}) in this equation?

Equation transcription:

Text transcription:

f{1}(x), f{2}(x), f{3}(x)

 f{1}(x) f{2}(x) f{3}(x) y f{1}^{prime}(x) f{2}(x) f{3}^{prime}(x) y f{1}^{prime}(x) f_{2}^{prime prime}(x) f{3}^{prime prime}(x) y^{prime prime} f{1}^{prime prime}(x) f{2}^{prime prime}(x) f{3}^{prime prime}(x) y^{prime prime}]=0

(f{1}, f{2}, f{3})

y^{prime prime}

ANSWER:

Solution

Step 1

In this problem, we have to find the coefficient of  in given equation.

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