Linear Second-Order ODEs
Second-Order Linear ODE
A second-order linear ODE has the general form
In our case, we consider only second-order ODE with constant coefficients:
In the special case where, we obtain the homogenous form
For the IVP, need two initial conditions:and
Linear Independence of Two Functions
Two functions
and are said to be linearly independent on the interval if and only if neither of them is a constant multiple of the other, for all .
A condition for Linear Independence of Solutions
Two solutions
and are linearly independent on the interval if
y_{1}‘(t) & y_{2}‘(t)\end{vmatrix}=y_{1}(t)y_{2}‘(t)-y_{1}‘(t)y_{2}(t)\neq 0$$
for some value
. If for all then and are linearly dependent.
Here,is called the Wronskian of .
General solution of Homogeneous second-order ODE
In general, if
and are two linearly independent solutions of the homogeneous second-order ODE then the general solution to this ODE is given by
for constants.
In the case the characteristic equationhas repeated roots then the general solution to the corresponding homogeneous 2nd-order ODE is given by
for constants.
Characteristic Equation with Complex Roots
Show that a solution of this equation has the general form
Sois indeed a soln to
Can you do the same for?
To showare linearly independent we calculate their Wronskian
-k\sin(kt) & k\cos(kt)\end{vmatrix}=(\cos(kt))(k\cos(kt))-(-k\sin(kt))(\sin(kt))=k\cos^2(kt)+k\sin^2(kt)=k\neq 0$$
Therefore,
will be our general soln to
Now let’s solve the same problem using its characteristic equation
Chara. Eqn.:
So according to Case 1, we do have 2 distinct soln.from the chara. eqn.
Euler's Formula
If
If