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'''Example:''' Consier the ODE <math>t^2 (lnt+1) \frac{d^2y}{dt^2} + t(2lnt+1) \frac{dy}{dt} -y=0</math>, one of the solutions is <math>y=\frac{1}{t}</math>. Find a second solution.
 
'''Example:''' Consier the ODE <math>t^2 (lnt+1) \frac{d^2y}{dt^2} + t(2lnt+1) \frac{dy}{dt} -y=0</math>, one of the solutions is <math>y=\frac{1}{t}</math>. Find a second solution.
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'''Solution:''' Try a general solution <math>y=\frac{x}{t}</math>, where <math>x=x(t)</math>, <math>y=y(t)</math>.
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Find the first and second derivative to <math>y</math>, hence <math>\frac{dy}{dt}=</math>
  
  

Revision as of 18:25, 22 November 2017

Advanced Methods to Solve 2nd-Order ODEs

A slecture by Yijia Wen

8.0 Abstract

Slightly we are moving to our last tutorial for the introduction of ordinary differential equations. This tutorial will give three particular methods to solve ODEs in the second order, which was also mentioned in 5.0. Theories are boring, we will be processing with sample questions.


8.1 Reduction of Order Method

This method is often used to find a second solution of an ODE, provided that one solution is given.

Example: Consier the ODE $ t^2 (lnt+1) \frac{d^2y}{dt^2} + t(2lnt+1) \frac{dy}{dt} -y=0 $, one of the solutions is $ y=\frac{1}{t} $. Find a second solution.


Solution: Try a general solution $ y=\frac{x}{t} $, where $ x=x(t) $, $ y=y(t) $.


Find the first and second derivative to $ y $, hence $ \frac{dy}{dt}= $


8.2 Cauchy-Euler Equation


8.3 Varation of Constant


8.4 Exercises


8.5 References

Institute of Natural and Mathematical Science, Massey University. (2017). 160.204 Differential Equations I: Course materials. Auckland, New Zealand.

Robinson, J. C. (2003). An introduction to ordinary differential equations. New York, NY., USA: Cambridge University Press.

Alumni Liaison

has a message for current ECE438 students.

Sean Hu, ECE PhD 2009