Advanced Methods to Solve 2nd-Order ODEs
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}=\frac{\frac{dx}{dt}}{t} - \frac{x}{t^2} $, $ \frac{d^2y}{dt^2}=\frac{2x}{t^3} - \frac{2\frac{dx}{dt}}{t^2} + \frac{\frac{d^2y}{dt^2}}{t} $.
Substitute them into ODE, hence $ t^2 (lnt+1) (\frac{2x}{t^3} - \frac{2\frac{dx}{dt}}{t^2} + \frac{\frac{d^2x}{dt^2}}{t}) + t (2lnt+1) (\frac{\frac{dx}{dt}}{t} - \frac{\frac{x}{t^2}}) - \frac{x}{t} = 0 $,
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.
