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Reduction of Order Method for Linear Homogeneous ODEs - Prof. Leslie M. Smith, Quizzes of Linear Algebra

University of Wisconsin (UW) - MadisonLinear Algebra
Prof. Leslie M. Smith

Instructions for using the reduction of order method to find linearly independent solutions for a linear homogeneous second-order ordinary differential equation. The method involves multiplying the given solution by a function v(x) and finding the derivative of v(x), followed by calculating the wronskian of the original solution and the modified solution.

Typology: Quizzes

2010/2011

Uploaded on 07/25/2011

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koofers-user-dgq 🇺🇸

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Math 320 (Smith): Optional Quiz 2
1. Given a solution y1(x) to the 2nd-order, linear, homogeneous ODE
y00(x) + p(x)y0(x) + q(x)y(x)=0
show that the method of Reduction of Order leads to a linearly independent solution.
We did this in class, but now try to reconstruct the argument on your own.
Step 1. Let y(x) = v(x)y1(x).
Step 2. Find v0(x).
Step 3. Find the Wronskian of y1(x) and y(x) = v(x)y1(x).
1

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Math 320 (Smith): Optional Quiz 2

  1. Given a solution y 1 (x) to the 2nd-order, linear, homogeneous ODE

y′′(x) + p(x)y′(x) + q(x)y(x) = 0

show that the method of Reduction of Order leads to a linearly independent solution. We did this in class, but now try to reconstruct the argument on your own. Step 1. Let y(x) = v(x)y 1 (x). Step 2. Find v′(x). Step 3. Find the Wronskian of y 1 (x) and y(x) = v(x)y 1 (x).