# Second Order Differential Equations

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A series of free Calculus 2 Video Lessons including examples and solutions. How to solve 2nd order differential equations?

Lecture 11: How to solve 2nd order differential equations.
A lecture on how to solve 2nd order (homogeneous) differential equations. The methods rely on the characteristic equation and the types of roots. Such ideas are seen in university and have numerous applications.

Solution to a 2nd order, linear homogeneous ODE with repeated roots
I discuss and solve a 2nd order ordinary differential equation that is linear, homogeneous and has constant coefficients. In particular, I solve y'' - 4y' + 4y = 0. The solution method involves reducing the analysis to the roots of of a quadratic (the characteristic equation). Such an example is seen in 1st and 2nd year university mathematics.

2nd order ODE with constant coefficients: simple method of solution
I give a simple and direct method to solve the ODE y'' + y' - 6y = 0. The method involves analysis of the associated characteristic equation. This type of problem is seen in 1st-year university mathematics.

2nd order ODE with constant coefficients: non-standard method of solution
I present a direct and non-standard method of solution to the 2nd order homogeneous ODE with constant coefficients, namely y'' - 4y' + 4y = 0. I do not explicitly use the characteristic equation, rather I reduce the problem to the analysis of a first order ODE. Such a method shows exactly why the solution features exponential functions and illustrates where they come from,

Lecture 12: How to solve second order differential equations.
A lecture on how to solve second order (inhomogeneous) differential equations. Plenty of examples are discussed and solved. The ideas are seen in university mathematics and have many applications to physics and engineering.

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