Khan Academy on a Stick
Second order linear equations
Linear differential equations that contain second derivatives

2nd Order Linear Homogeneous Differential Equations 1
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Introduction to 2nd order, linear, homogeneous differential equations with constant coefficients.

2nd Order Linear Homogeneous Differential Equations 2
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Let's find the general solution!

2nd Order Linear Homogeneous Differential Equations 3
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Let's use some initial conditions to solve for the particular solution

2nd Order Linear Homogeneous Differential Equations 4
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Another example with initial conditions!
Linear homogeneous equations
To make your life interesting, we'll now use the word "homogeneous" in a way that is not connected to the way we used the term when talking about firstorder equations. As you'll see, second order linear homogeneous equations can be solved with a little bit of algebra (and a lot of love).

Complex roots of the characteristic equations 1
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What happens when the characteristic equations has complex roots?!

Complex roots of the characteristic equations 2
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What happens when the characteristic equation has complex roots?

Complex roots of the characteristic equations 3
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Lets do an example with initial conditions!

Repeated roots of the characteristic equation
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What happens when the characteristic equation only has 1 repeated root?

Repeated roots of the characteristic equations part 2
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An example where we use initial conditions to solve a repeatedroots differential equation.
Complex and repeated roots of characteristic equation
Thinking about what happens when you have complex or repeated roots for your characteristic equation.

Undetermined Coefficients 1
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Using the method of undetermined coefficients to solve nonhomogeneous linear differential equations.

Undetermined Coefficients 2
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Another example using undetermined coefficients.

Undetermined Coefficients 3
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Another example where the nonhomogeneous part is a polynomial

Undetermined Coefficients 4
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Putting it all together!
Method of undetermined coefficients
Now we can apply some of our second order linear differential equations skills to nonhomogeneous equations. Yay!