Fundamental Solutions
Theorem :
If the functions are solutions to the
Equation 1 with and their Wronskian
..
..
121
121
−
−
nn
nn
yyyy
yyyy
)(),(),...,(),(121 tytytyty nn−
0)( =tg
General Solution to Nonhomogeneous ODE
Theorem:
The general solution to the nth order linear nonhomogeneous ODE
Equation 1 can be written as
Particular Solution to
Nonhomogeneous ODE
Theorem
The solution to Equation 1 with nonzero and zero initial
conditions is
)(tg
Higher Order Linear Time-Invariant ODEs
Consider the ODE
0…
1
1
1=+++ −
−
ya
dt
yd
a
dt
yd
n
n
n
n
n
Roots of the Characteristic Equation and
Corresponding Solutions
If the root appears only once it represents the solution
If the root appears m times it represents the solution
1
rr =
tr
ecty 1
1
)( =
Initial Conditions and Coefficients
Coefficients are determined from equations for the initial conditions:
n linear equations in n unknowns, providing a unique solution
because the determinant of the equations (the Wronskian) is
nonzero :
00022011
)(…)()(
=+++
nn
ytyctyctyc
Solving Linear Nonhomogeneous ODEs:
Undetermined Coefficient Approach
The easiest way to solve nth order nonhomogeneous ODEs is often
by guessing the form of a solution based on the input . The
approach is effective for most interesting forms of .
Example:
To solve
)(tg
)(tg
tya
dt
dy
a
dt
yd
a
dt
yd
a
dt
yd
sin
43
2
2
2
3
3
1
4
4
=++++
Solving Linear Inhomogeneous ODEs:
Undetermined Coefficient Approach (2)
We must have
0)()( 3
3
14
2
2
4=−−+− DaaCaa
Higher Order Linear ODEs
Engineering Example:
Satellite Orbit Stability
Satellite Orbit Stability
Consider a satellite in low earth orbit (200 – 400 miles altitude) . The
principal force acting on it is gravity but though the density of air
is very low the long-term effect of air resistance (drag) cannot be
ignored.
y
x
22
2/322
2)(
yx
m
xSC
yx
GMx
x
D
+−
+
−=
Satellite Orbit Stability (2)
In cartesian coordinates the
satellite’s equations of motion are