Khan Academy on a Stick
Second order linear equations
Linear differential equations that contain second derivatives
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2nd Order Linear Homogeneous Differential Equations 1
ccIntroduction to 2nd order, linear, homogeneous differential equations with constant coefficients.
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2nd Order Linear Homogeneous Differential Equations 2
ccLet's find the general solution!
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2nd Order Linear Homogeneous Differential Equations 3
ccLet's use some initial conditions to solve for the particular solution
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2nd Order Linear Homogeneous Differential Equations 4
ccAnother 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 first-order equations. As you'll see, second order linear homogeneous equations can be solved with a little bit of algebra (and a lot of love).
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Complex roots of the characteristic equations 1
ccWhat happens when the characteristic equations has complex roots?!
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Complex roots of the characteristic equations 2
ccWhat happens when the characteristic equation has complex roots?
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Complex roots of the characteristic equations 3
ccLets do an example with initial conditions!
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Repeated roots of the characteristic equation
ccWhat happens when the characteristic equation only has 1 repeated root?
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Repeated roots of the characteristic equations part 2
ccAn example where we use initial conditions to solve a repeated-roots differential equation.
Complex and repeated roots of characteristic equation
Thinking about what happens when you have complex or repeated roots for your characteristic equation.
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Undetermined Coefficients 1
ccUsing the method of undetermined coefficients to solve nonhomogeneous linear differential equations.
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Undetermined Coefficients 2
ccAnother example using undetermined coefficients.
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Undetermined Coefficients 3
ccAnother example where the nonhomogeneous part is a polynomial
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Undetermined Coefficients 4
ccPutting it all together!
Method of undetermined coefficients
Now we can apply some of our second order linear differential equations skills to nonhomogeneous equations. Yay!