"How Three Functions of Wroian are used to..." solved and explained

College
Calculus and Analysis
Differential Calculus
Analyzing functions

Susan Munoz

Answered question

2021-11-22

How Three Functions of Wro
ian are used to Find the Homogeneous Linear Differential Equation Whose Solution Space is Spanned by Two Given Functions?

Answer & Explanation

Prioned

Beginner2021-11-23Added 11 answers

Step1
Let's consider two function y;(x) and y,(x) are the linearly independent solution of
$y p (x) y^{'} + q (x) y = 0$
Let y(x) is the general solution of this equation.
Thus, $y (x) = C_{1} y_{1} (x) + C_{2} y_{2} (x)$ , where $C_{1}$ and $C_{2}$ ,are constants.
Step 2
Now by calculating the Wro
ian
$W [y_{1}, y_{2}, y] = 0$
$W [y_{1}, y_{2}, y] = y " + p (x) y^{'} + q (x) y$
Where,
$p (x) = - \frac{W^{'} (x)}{W (x)}$
$q (x) = ∣ \frac{W (y_{1}^{'}, y_{2}^{'})}{W (y_{1}, y_{2})}$

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