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Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16 Second Order Differential Equations
Linear Equations. We brieﬂy recall some facts about diﬀerential equations which we have seen over the
past few weeks. Recall that we use the notation
dn y
y ( n) = n
dt
to denote the nth derivative. We say that an equation of the form
F t, y, y (1) , . . . , y (n) = 0 is an nth order diﬀerential equation. In particular, we can express the highest order derivative in terms of
the lower order derivatives:
dn y
= G t, y, y (1) , . . . , y (n−1)
dtn
for some function G(t, y1 , y2 , . . . , yn ). When n = 2, we call an equation of the form
d2 y
dy
= G t, y,
dt2
dt
a second order diﬀerential equation.
Recall that a ﬁrst order diﬀerential equation is said to be a linear equation if it is in the form
dy
= G(t, y )
where
G(t, y ) = g (t) − p(t) y
dt
for some functions p(t) and g (t). Similarly, we say that a second order diﬀerential equation is a linear
equation if it is in the form
d2 y
dy
dy
= G t, y,
where
G (t, y1 , y2 ) = g (t) − q (t) y − p(t)
dt2
dt
dt
for some functions p(t), q (t), and g (t). Any second order diﬀerential equation which cannot be placed in this
form is called a nonlinear equation. Equivalently, we say that a second order diﬀerential equation is linear if
it is in the form
d2 y
dy
+ p(t)
+ q (t) y = g (t).
2
dt
dt
We mention in passing that sometimes we consider second order diﬀerential equations in the form
a(t) y + b(t) y + c(t) y = f (t).
Upon dividing both sides by a(t), we can place this equation in the one above, where
p(t) = b(t)
,
a(t) q (t) = c(t)
,
a(t) and g (t) = f (t)
.
a(t) Example. Consider an object of mass m which is under the inﬂuence of gravity and air resistance. Let
x = x(t) denote the height of the mass at time t. The force due to gravity is m g , and the force due to air
resistance is γ v for some constant γ . Newton’s Second Law of Motion states that F = m a is the sum of the
forces on the object. This gives the diﬀerential equation
m d2 x
dx
= −m g − γ
.
dt2
dt Hence we have an equation in the form
d2 x
dx
= G t, x,
where
dt2
dt
This is a linear second order diﬀerential equation. 1 G(t, x1 , x2 ) = −g − γ
x2 .
m ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins
 Differential Equations, Linear Equations, Equations

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