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
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
Theorem
Consider the differential equation
L[y ] = y 00 (t) + p(t)y 0 (t) + q(t)y (t) = 0
whose coefficients p and q are continuous on some open interval I.
Choose some point t0 in I. Let y1 and y2 be the solutions of the
ODE that also satisfy the initial conditions
y (t0 ) = 1,
y 0 (t0 ) = 0;
y (t0 ) = 0, y 0 (t0 ) = 1
respectively. Then y1 and y2 form a fundamental set of solutions of
the ODE.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
Example 39
Find the fundamental set of solutions y1 and y2 specified by the
previous Theorem for the differential equation
y 00 (t) − y (t) = 0
using the initial point t0 = 0.
Solution
Let us denote by y1 (t) the solution of the equation that satisfies
the initial conditions
y (t0 ) = 1,
Dr. Marco A Roque Sol
y 0 (t0 ) = 0
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
The general solution is
y = c1 e t + c2 e −t
and the initial conditions are satisfied if c1 = 1/2 and c2 = 1/2.
Thus
1
1
y1 = e t + e −t = cosh(t)
2
2
In the same way, if y2 (t) is the solution of the equation that
satisfies the initial conditions
y (t0 ) = 0,
Dr. Marco A Roque Sol
y 0 (t0 ) = 1
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
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Wrosnkian
then
1
1
y2 = e t − e −t = sinh(t)
2
2
Since the Wronskian of y1 and y2 is
W = cosh2 (t) − sinh2 (t) = 1
these functions also form a fundamental set of solutions, as stated
in the last Theorem. Therefore, the general solution of the
equation y 00 (t) − y (t) = 0 can be written as
y = k1 cosh(t) + k2 sinh(t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
In the next section we will encounter equations that have
complex-valued solutions. The following theorem is fundamental in
dealing with such equations and their solutions.
Theorem
Consider again the equation
L[y ] = y 00 (t) + p(t)y 0 (t) + q(t)y (t) = 0
where p and q are continuous real-valued functions. If
y = u(t) + iv (t) is a complex-valued solution of the above
equation, then its real part u and its imaginary part v are also
solutions of this equation.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
Proof
To prove this theorem we substitute u(t) + iv (t) for y in L[y ],
obtaining
L[y ] = (u(t) + iv (t))00 (t) + p(t)(u(t) + iv (t))0 (t) + ....
... + q(t)(u(t) + iv (t))
L[y ] = (u 00 (t) + iv 00 (t))(t) + p(t)(u 0 (t) + iv 0 (t))(t) + ...
... + q(t)(u(t) + iv (t))
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
L[y ] = (u(t)00 + p(t)u(t) + q(t)u(t)) + i(v (t)00 (t) + v (t)0 (t) + v (t))
L[y ] = L[u](t) + iL[v ](t) = 0
.
Recall that a complex number is zero if and only if its real and
imaginary parts are both zero. Therefore, L[u](t) = 0 and
L[v ](t) = 0 also; consequently, u and v are also solutions of the
ODE.
OBS If y is a complex-valued solution of the ODE
L[y ] = y 00 (t) + p(t)y 0 (t) + q(t)y (t) = 0
with p(t) and q(t) real-valued function, the ȳ (t) is also a solution.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
The following theorem, gives a simple explicit formula for the
Wronskian of any two solutions of any linear second order
differential equation, even if the solutions themselves are not
known.
Abel’s Theorem
(This result was derived by the Norwegian mathematician Niels
Henrik Abel (18021829) in 1827 and is known as Abel’s formula.
https://en.wikipedia.org/wiki/Niels_Henrik_Abel ).
If y1 and y2 are solutions of the differential equation
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
L[y ] = y 00 (t) + p(t)y 0 (t) + q(t)y (t) = 0
where p and q are continuous on an open interval I,
then the Wronskian W (y1 , y2 )(t) is given by
W (y1 , y2 )(t) = ce −
R
p(t)dt
where c is a certain constant that depends on y1 and y2 , but not
on t. Moreover, W (y1 , y2 )(t) either is zero for all t in I (if c = 0)
or else is never zero in I (if c 6= 0).
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
Proof
We start by noting that y1 and y2 satisfy
L[y1 ] = y100 (t) + p(t)y10 (t) + q(t)y1 (t) = 0
L[y2 ] = y200 (t) + p(t)y20 (t) + q(t)y2 (t) = 0
If we multiply the first equation by −y 2
−y2 y100 (t) − p(t)y2 y10 (t) − q(t)y2 y1 (t) = 0
multiply the second by y1
y1 y200 (t) + p(t)y1 y20 (t) + q(t)y1 y2 (t) = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Solutions of Linear Homogeneus Systems. The
Wrosnkian
and add the resulting equations, we obtain
(y1 y200 (t) − y2 y100 (t)) + p(t)(y1 y20 (t) − y10 y2 (t)) = 0
Next, we let W (y1 , y2 )(t) and observe that
W (y1 , y2 )0 (t) = y1 y200 (t) − y100 y2 (t)
Then we have
W 0 + p(t)W = 0
and the solution is
W (y1 , y2 )(t) = ce −
Dr. Marco A Roque Sol
R
p(t)dt
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
However, since the exponential function is never zero, W (t) is not
zero unless c = 0, in which case W (t) is zero for all t
Let’s continue our discussion of the equation
ay 00 + by 0 + cy = 0
where a, b, and c are given real numbers. We found that if we seek
solutions of the form y = e rt , then r must be a root of the
characteristic equation
ar 2 + br + c = 0
Now, consider the case where the roots are conjugate complex,
that is, b 2 − 4ac < 0. In this case we denote them by
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
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Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
r1 = λ + iµ;
r2 = λ − iµ
where λ and µ are real numbers. The corresponding expressions
for the two possible solutions are
y1 (t) = e r1 t = e (λ+iµ)t ;
y2 (t) = e r2 t = e (λ−iµ)t
and we can show that W (y1 , y2 )(t) 6= 0 therefore the general
solution is
y (t) = c1 y1 (t) + c2 y2 (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Second Order Differential Equations
Complex Roots of the Characteristic Equation
OBS
1.- If z(t) = f (t) + ig (t) is a complex-valued function, where f (t)
and g (t) are real-valued function and differentiable on some
interval I, then z is differentiable on I and z 0 (t) = f 0 (t) + ig 0 (t)
2.- Using a Taylor expansion for the complex exponential
e
irt
∞
X
(irt)n
=
e irt =
n=0
∞
X
n=0
e
irt
=
∞
X
n=0,2,4,...
(i)n (rt)n
+i
n!
n!
(rt)n i n
n!
∞
X
(i)n
n=1,3,5,...
Dr. Marco A Roque Sol
(rt)n
;
n!
i n = 1, i, −1, −i, ...
Ordinary Differential Equations
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e irt =
∞
X
(−1)n (rt)2n
(2n)!
n=0
+i
∞
X
(rt)2n+1
(−1)n+1
(2n + 1)!
n=0
but
cos(rt) =
∞
X
(−1)n (rt)2n
n=0
(2n)!
;
and
sin(rt) =
∞
X
(−1)n+1 (rt)2n+1
n=0
(2n + 1)!
Therefore
e irt = cos(rt) + i sin(rt)
The above equation is known as Eulers formula after the Swiss
Mathematician/Physicist Leonhard Euler.
https:/en.wikipedia.org/wiki/Leonhard_Euler
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
Example 40
Solve the IVP
y 00 + y 0 + 9.25y = 0;
y (0) = 2, y 0 (0) = 8
Solution
The characteristic equation is
r 2 + r + 9.25 = 0
so its roots are
1
r1 = − + 3i;
2
Dr. Marco A Roque Sol
1
r2 = − − 3i
2
Ordinary Differential Equations
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Complex Roots of the Characteristic Equation
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Complex Roots of the Characteristic Equation
In this case the Wronskian is given by W (y1 , y2 )(t) = −6ie −t
which is not zero, so the general can be expressed as a linear
combination of y1 (t) and y2 (t) with arbitrary coefficients.
Now, by the principle of superposition, we know that any linear (
real or complex ) combination of two solutions is also a solution of
the differential equations, in particular
1
(y1 + y2 );
2
1
(y1 − y2 )
2i
are solutions of the differential equation
1
1
1
1
(y1 + y2 ) = (e (− 2 +3i)t + e (− 2 −3i)t )
2
2
t
t
1
1
(y1 + y2 ) = (e − 2 e 3ti + e − 2 e −3ti )
2
2
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
1
1
(y1 + y2 ) = [e −t/2 (cos(3t) + i sin(3t)) + e −t/2 (cos(3t) − i sin(3t))]
2
2
1
(y1 + y2 ) = e −t/2 cos(3t)
2
In a similar way we find
1
(y1 − y2 ) = e −t/2 sin(3t)
2i
Thus, in the case of comjugate complex roots
r1 = λ + iµ;
Dr. Marco A Roque Sol
r2 = λ − iµ,
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
the general solution of the linear second order homogeneus
equation
ay 00 + by 0 + cy = 0
will be written in the form
y (t) = c1 e λt cos(µt) + c2 e λt sin(µt)
Example 41
Find the particular real-valued solution of the IVP
y 00 + y 0 + 9.25y = 0;
y (0) = 2, y 0 (0) = 8
Solution
We found that the roots are r1 = − 21 + 3 i;
Dr. Marco A Roque Sol
r2 = − 12 − 3 i
Ordinary Differential Equations
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Complex Roots of the Characteristic Equation
Therefore the general solution is written as
1
1
y (t) = c1 e − 2 t cos(3t) + c2 e − 2 t sin(3t)
and
1
1
1
y 0 (t) = − c1 e − 2 t cos(3t) − 3c1 e − 2 t sin(3t)...
2
1
1
1
... − c2 e − 2 t sin(3t) + 3c2 e − 2 t cos(3t)
2
applying initial conditions, we obtain
c1 = 2;
1
− c1 + c2 = 8
2
c1 = 2;
Dr. Marco A Roque Sol
c2 = 3
Ordinary Differential Equations
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therefore the solution is
t
1
t
y (t) = 2e − 2 cos(3t) + 3e − 2 t sin(3t) = e − 2 (2cos(3t) + 3sin(3t))
The graph of this solution is shown below
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
Example 42
Find the particular real-valued solution of the IVP
16y 00 − 8y 0 + 145y = 0;
y (0) = −2, y 0 (0) = 1
Solution
The characteristic equation is 16r 2 8r + 145 = 0 and its roots are
r = 41 ± 3 i. Thus the general solution of the differential equation
and its derivative are
t
t
y (t) = c1 e 4 cos(3t) + c2 e 4 sin(3t)
Dr. Marco A Roque Sol
Ordinary Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Second Order Differential Equations
Complex Roots of the Characteristic Equation
t
t
1
y 0 (t) = c1 e 4 cos(3t) − 3c2 e 4 sin(3t)...
4
t
t
1
... + c1 e 4 cos(3t) + 3c2 e 4 cos(3t)
4
applying initial conditions, we obtain
y (0) = c1 = −2;
1
y 0 (0) = c1 + 3c2 = 1
4
c1 = −2;
Dr. Marco A Roque Sol
c2 =
1
2
Ordinary Differential Equations
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therefore the solution and its graph are
t
1 t
y (t) = −2e 4 cos(3t) + e 4 sin(3t)
2
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
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Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
Example 43 (Harmonic Oscillator)
Find the particular real-valued solution of ODE
y 00 + ω 2 y = 0;
y (0) = y0 , y 0 (0) = v0
Solution
The characteristic equation is r 2 + ω 2 = 0 and its roots are
r = ±iω. Thus the general solution of the differential equation and
its derivative are
y (t) = c1 cos(ωt) + c2 sin(ωt)
y 0 (t) = −ωc1 sin(ωt) + ωc2 cos(ωt)
Dr. Marco A Roque Sol
Ordinary Differential Equations
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and applying initial conditions
c1 = y0
c2 =
v0
ω
and the solution is
y (t) = y0 cos(ωt) +
v0
sin(ωt)
ω
As we can see, in this case the solution is purely oscillatory. For
different initial positions y0 and velocities v0 we have the following
graph
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Complex Roots of the Characteristic Equation
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Repeated Roots. Reduction of Order
In this case we want solutions to
ay 00 + by 0 + cy = 0
where solutions to the characteristic equation
ar 2 + br + c = 0
are double roots (b 2 − 4ac = 0) r1 = r2 = −b/2a. This leads to
a problem however. Recall that the solutions are
bt
y1 (t) = e r1 t = e − 2a
and
bt
y2 (t) = e r2 t = e − 2a
These are the same solution. We can use the first solution, but
we’re going to need a second solution. We propose a solution of
the form y2 (t) = ν(t)y1 (t).
Dr. Marco A Roque Sol
Ordinary Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Second Order Differential Equations
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where ν(t) is a function that needs to be determined, if it is
possible. To determine if this in fact can be done, let’s plug this
back into the differential equation and see what we get
ay 00 + by 0 + cy = a(ν(t)y1 (t))00 + b(ν(t)y1 (t))0 + c(ν(t)y1 (t)) = 0
bt
but since y (t) = ν(t)e − 2a
bt
y 0 = ν 0 (t)e − 2a − ν(t)
b − bt
e 2a
2a
bt
bt
b bt
b2
y 00 = ν 00 (t)e − 2a − e − 2a ν 0 (t) + 2 e − 2a ν(t)
a
4a
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Repeated Roots. Reduction of Order
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Then, by substituting in the ODE, we obtain
bt
bt
b bt
b2
a(ν 00 (t)e − 2a − e − 2a ν 0 (t) + 2 e − 2a ν(t)) + ...
a
4a
bt
b(ν 0 (t)e − 2a − ν(t)
bt
b − bt
e 2a ) + c(ν(t)e − 2a ) = 0
2a
Canceling the factor e −bt/2a , which is nonzero, and rearranging the
remaining terms, we find that
aν 00 (t) + (−b + b)ν 0 (t) + (
Dr. Marco A Roque Sol
b2 b2
−
+ c)ν(t)) = 0
4a 2a
Ordinary Differential Equations
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and taking in account that b 2 − 4ac = 0, then
ν 00 (t) = 0
ν(t) = k1 + k2 t
Hence, we have that the general solution is
bt
bt
y (t) = C1 e 2a + (k1 + k2 t)e 2a
bt
bt
y (t) = c1 e 2a + c2 te 2a
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Repeated Roots. Reduction of Order
Thus y is a linear combination of the two solutions
bt
y1 (t) = e 2a
and
bt
y2 = te 2a
The Wronskian of these two solutions is
W (y1 , y2 )(t) = e
bt
a
which is never zero, therefore the solutions y1 and y2 , given above,
are a fundamental set of solutions.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Second Order Differential Equations
Solutions of Linear Homogeneus Systems. The Wrosnkian
Complex Roots of the Characteristic Equation
Repeated Roots. Reduction of Order
Repeated Roots. Reduction of Order
Example 44
Find the solution of the initial value problem
y 00 − y 0 + 0.25y = 0;
y (0) = 0, y 0 (0) =
1
3
Solution
The characteristic equation is
r 2 − r + 0.25 = 0
and its roots are r = r1 = r2 = 21 . Thus the general solution of
the differential equation and its derivative are
y (t) = c1 e t/2 + c2 te t/2
1
1
y 0 (t) = c1 e t/2 + c2 e t/2 + c2 te t/2
2
2
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Applying initial conditions
2 = y (0) = c1
1
1
= y 0 (0) = c1 + c2 t
3
2
so c1 = 2 and c2 = 2/3. Thus the solution of the initial value
problem is
2
y (t) = 2e t/2 − te t/2
3
Dr. Marco A Roque Sol
Ordinary Differential Equations
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The graph of this solution is shown below
Dr. Marco A Roque Sol
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