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Unformatted text preview: 2 MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16 More Deﬁnitions. Consider a linear second order diﬀerential equation
a(t) y + b(t) y + c(t) y = f (t).
If we have initial conditions in the form
dy
(t0 ) = y0 ;
dt
we call this system an initial value problem. Note that an initial value problem speciﬁes
(1) an initial point (t0 , y0 ), and
(2) an initial slope y0 at that point.
We say that the diﬀerential equation is homogeneous if f (t) = 0 for all t i.e., if the diﬀerential equation
is in the form
a(t) y + b(t) y + c(t) y = 0.
y (t0 ) = y0 and The functions a(t), b(t), and c(t) are called the coeﬃcients of the diﬀerential equation. Any second order
diﬀerential equation which cannot be placed in this form is said to be nonhomogeneous equation or an
inhomogeneous equation.
Constant Coeﬃcient Equations. Say for the moment that the coeﬃcients of the diﬀerential equation
are constant functions:
a(t) = a,
b(t) = b,
and
c(t) = c.
We consider equations in the form
a y + b y + c y = f (t).
Say for the moment that this is a homogeneous equation. We give a trick to solve such an equation.
For inspiration, consider the analogue of a constant coeﬃcient homogeneous equation of the ﬁrst order:
a y + b y = 0.
We can solve this equation because it is separable:
a y = −b y 1 dy
b
=−
y dt
a
d
b
ln y  = −
dt
a
b
ln y  = − t + (constant)
=⇒
y (t) = (constant) · ert
a
in terms of the constant
b
r=−
=⇒
a r + b = 0.
a
Hence the solution to the diﬀerential equation a y + b y = 0 is essentially the exponential y = ert for some
constant r. Note that the equation a r + b = 0 is similar in form to the diﬀerential equation.
Now we return to the constant coeﬃcient homogeneous equation of the second order. We will guess that
the solution is y = ert for some constant r. We have the derivatives
y (t) = ert , y (t) = r ert , and y (t) = r2 ert . This gives the relation
0 = a y + b y + c y = a r2 ert + b r ert + c ert = a r2 + b r + c ert . The exponential ert = 0, so we must have a r2 + b r + c = 0. This equation is called the characteristic equation of the homogeneous diﬀerential equation.
We will show the following. Consider the constant coeﬃcient homogeneous equation
a y + b y + c = 0. ...
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 Spring '09
 EdrayGoins

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