Second Order Differential Equations
Ordinary Differential Equations
Dr. Marco A Roque Sol
12/01/2015
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Repeated Roots. Reduction of Order
Reduction of order
The procedure used in this section for equations with constant
coefficients is more generally applicable. Suppose that we know
one solution y1 (t), not everywhere zero, of
y 00 + p(t)y 0 + q(t)y = 0
A second solution can be proposed as y = v (t)y1 (t)
then
y 0 = v 0 (t)y1 (t) + v (t)y10 (t)
y 00 = v 00 (t)y1 (t) + v 0 (t)y10 (t) + v (t)y100 (t) + v 0 (t)y10 (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Repeated Roots. Reduction of Order
Substituting for y , y 0 , and y 00 in the ODE and rearrenging the
terms we find
y 00 + py 0 + qy = (v 00 (t)y1 (t) + v 0 (t)y10 (t) + v (t)y100 (t) + v 0 (t)y10 (t)) + ...
... + p(v 0 (t)y1 (t) + v (t)y10 (t)) + q(v (t)y1 (t)) = 0
=⇒ y1 v 00 + (2y10 + y1 )v 0 + (y100 + py10 + qy1 )v = 0
Since y1 is a solution of the differential equation, the coefficient of
v in the above equation is zero, therefore we have
y1 v 00 + (2y10 + y1 )v 0 = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Repeated Roots. Reduction of Order
The method has the name of reduction of order, because is
actually a first order equation for the function v 0 and can be solved
either as a first order linear equation or as a separable equation.
Once v 0 has been found, then v is obtained by an integration.
Example 45
Given that y1 (t) = t −1 is a solution of
2t 2 y 00 + 3ty 0 − y = 0;
t>0
Find another solution y2
Solution
We set y = v (t)t −1 , then
y 0 = v 0 (t)t −1 − v (t)t −2 ;
y 00 = v 00 (t)t −1 − 2v 0 (t)t −2 + 2v (t)t −3
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Repeated Roots. Reduction of Order
Substituting for y , y 0 and y 00 in the differential equation and
collecting terms, we obtain
2t 2 (v 00 (t)t −1 − 2v 0 (t)t −2 + 2v (t)t −3 ) + ...
... + 3t(v 0 (t)t −1 − v (t)t −2 ) − (v (t)t −1 ) = 0
2tv 00 (t) + (−4 + 3)v 0 (t) − (4t −1 ) − 3t −1 − t −1 )v (t) = 0
2tv 00 (t) − v 0 (t) = 0
If we let w = v , then the above equation becomes
2tw 0 (t) − w (t) = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Repeated Roots. Reduction of Order
Separating the variables and solving for w (t), we find that
w (t) = ct 1/2 , then
2
v (t) = ct 3/2 + k
3
It follows that
2
y = v (t)t −1 = ct 1/2 + kt −1
3
where c and k are arbitrary constants. The second term on the
equation is a multiple of y1 (t) and can be dropped, but the first
term provides a new solution y2 (t) = t 1/2 In this case the
Wronskian of y1 (t) and y2 (t) is given by
3
W (y1 , y2 ) = t −3/2 6= 0
2
Consequently, y1 and y2 form a fundamental set of solutions of the
ODE for t > 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
We now return to the nonhomogeneous equation
L[y ] = y 00 + p(t)y 0 + q(t)y = g (t)
where p, q, and g are continuous functions on the open interval I.
The equation with g (t) = 0
L[y ] = y 00 + p(t)y 0 + q(t)y = 0
is called the associate homogeneous equation .
Theorem
If Y1 and Y2 are two solutions of the nonhomogeneous equation,
then their difference Y1 − Y2 is a solution of the corresponding
homogeneous equation. If, in addition, y1 and y2 are a fundamental
set of solutions of the associte homogeneous equation, then
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
Y1 (t) − Y2 (t) = c1 y1 (t) + c2 y2 (t),
where c1 and c2 are certain constants.
Theorem
The general solution of the nonhomogeneous equation
L[y ] = y 00 + p(t)y 0 + q(t)y = g (t)
can be written in the form
y = φ(t) = c1 y1 (t) + c2 y2 (t) + YP (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
where y1 and y2 are a fundamental set of solutions of the
corresponding associate homogeneous equation, c1 and c2 are
arbitrary constants, and YP is some specific solution of the
nonhomogeneous equation.
In other words, if we want to solve the nonhomogeneous equation
what we need to do is
1. Find the general solution c1 y 1(t) + c2 y2 (t) of the corresponding
homogeneous equation. This solution is frequently called the
complementary solution and may be denoted by yc (t).
2. Find some single solution YP (t) of the nonhomogeneous
equation. Often this solution is referred to as a particular solution.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
3. Form the sum of the functions found in steps 1 and 2.
Thus, we need to find a method to find a particular solution of the
nonhomogeneus case. In this direction we introduce the
Method of Undetermined Coefficients
The method is quite simple. All that we need to do is look at g (t)
and make a guess as to the form of YP (t) leaving the coefficient(s)
undetermined (and hence the name of the method). Plug the
guess into the differential equation and see if we can determine
values of the coefficients. If so, we guessed correctly, otherwise we
guessed incorrectly.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
OBS
There are two disadvantages to this method
1) It will only work for a fairly small class of g (t)0 s.
2) It is generally only useful for constant coefficient differential
equations.
Example 46
Determine a particular solution to
y 00 − 4y 0 − 12y = 3e 5t
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
Solution
We need to make a guess as to the form of a particular solution to
this differential equation. Since g (t) is an exponential and the
coefficients are constants, it seems that a likely form of the
particular solution would be
YP = Ae 5t
What we need to do is do a couple of derivatives, plug this into the
differential equation and see if we can determine the value for A .
y 00 − 4y 0 − 12y = e 5t
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
y 00 − 4y 0 − 12y = (Ae 5t )00 − 4(Ae 5t )0 − 12(Ae 5t ) = 3e 5t
y 00 − 4y 0 − 12y = 25Ae 5t − 4(5Ae 5t ) − 12Ae 5t = 3e 5t
y 00 − 4y 0 − 12y = −7Ae 5t = 3e 5t
so, the value for the undetermined coefficient is A = −3/7 and the
particular solution is
3
YP = − e 5t
7
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
Example 47
Determine the solution to the IVP
18 0
1
, y (0) = −
7
7
We know that the general solution will be of the form,
y 00 − 4y 0 − 12y = 3e 5t : y (0) =
3
y (t) = c1 y1 + c2 y2 + YP = c1 e −2t + c2 e 6t − e 5t
7
and its derivative is
y 0 (t) = −2c1 e −2t + 6c2 e 6t −
Dr. Marco A Roque Sol
15 5t
e
7
Ordinary Differential Equations
Second Order Differential Equations
Repeated Roots. Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficie
Nonhomogeneous Equations; Method of
Undetermined Coefficients
Now, apply the initial conditions to these.
18
3
= y (0) = c1 + c2 −
7
7
1
15
− = y 0 (0) = −2c1 + 6c2 −
7
7
Solving this system gives c1 = 2 and c2 = 1. The actual solution is
then
y (t) = 2e −2t + e 6t −
Dr. Marco A Roque Sol
3
7e 5t
Ordinary Differential Equations