1. Linear Equations Constant Coefficients
2nd Order Differential Equations

2. Homogeneous Linear Equations
Linearly independent solutions (guaranteed 2)
General solutions (with 2 constants)
Particular solutions (with 2 IC)
Wronskian for linear independence
Next - Characteristic equation for linear equations with constant coefficients
Simple real roots
Repeated real roots
Complex roots

3. Linear 2nd Order Equations with constant coefficients
General form:
At careful look, a solution y(x) will have the property that its derivatives are constant multiples of itself.
We have a class of functions like that, namely
Where r is just a constant (number). Then

4. Let’s look at a specific example
Suppose that
Assume a solution of the form
Work as far as you can with this y, see what happens

5. In general Suppose (no IC for now) Assume a solution
Compute the equation with this as y to get
Or, (*)
Since there is no way for
(*) will = 0 exactly when
This is a quadratic which can be solved for r using methods from HS algebra.

6. “Trick” for these equations
The equation is called the characteristic equation of the DE
The solutions of the DE come from the roots of the characteristic equation

7. Complex roots of the characteristic eq
Requires a little complex analysis (not here though)
Recall that complex roots come in pairs
The solns of our equation can be read off of the roots.

8. Example
Find a general solution to the DE

9. IVPs
Find a solution to the IVP