Linear Equations Constant Coefficients 2nd Order Differential Equations

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

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

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

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.

“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

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.

Example Find a general solution to the DE

IVPs Find a solution to the IVP