Introduction to Differential

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Introduction to Differential Equations
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Warmup At each point (x0,y0) on a function y
f(x), the tangent line has the equation y 2x0x -
y0. Find the equation for f(x). Solution The
equation for the tangent line to y f(x) at
(x0,y0) is y y0 f(x0)(x x0)
OR y f(x0)x (y0 x0f(x0)) But we are
given that the equation of the tangent is y 2x0x
y0. Hence, f(x0) 2x0 and y0 x0f(x0) y0
which gives y0 x02. Thus, f(x) x2.
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• We begin with two examples.
• Find any function whose derivative is
• f(x) 9x24x b) f(x)3cos2x

F(x) 3x3 2x2
F(x) 1.5sin2x
These are the antiderivatives
An antiderivative of the function f(x) is a
function F(x) that satisfies F(x) f(x)
How many possible antiderivatives are there for
each function?
- There are an infinite number of solutions!
- They can be represented as F(x) 3x3 2x2C
F(x) 1.5sin2xC
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Solve
with the initial condition s 3 when t 0.
Solution S(t) t2 C is the most general
antiderivative of 2t. At s 3 and t 0
3 02 C Hence C 3 Therefore S(t)
t2 3
Solve
when y 1 and t 2.
Solution X(t)
at y 1 and t 2
C
Therefore X(t)
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Determine the equation of the curve y f(x) that
passes through (0, 1) and satisfies
Solution y 2x2 x C at (0, 1), C
1 Hence y 2x2 x 1
Note (1) the graphs are parallel since the
constant distance between points with the same
x-coordinate (2) the set of all solutions of the
form y F(x) C is a one-parameter family of
solution curves, with C being the parameter
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• Find the antiderivative of each
• a) f(x) sin px
• b) f(x) (2x5)4
• c)

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Many of the general laws of nature find their
most useful form in equations that involve rates
of change. These equations are called
differential equations because they contain
functions and their differential quotients. Some
examples of differential equations are
P 3P
We have begun by working with equations of the
form y f(x), the solutions of which are
called antiderivatives. Recall an
antiderivative of a function f(x) is a function
F(x) where F(x) f(x). This is a simple case
of a d.e. More generally, a differential
equation is any equation that involves an
unknown Function and its derivatives. For
example,
Where k is a constant is a common form of a d.e.,
with y denoting the unknown function. The
process of finding the unknown function is
referred to as solving the d.e.. Any function
that when substituted for the unknown function,
reduces the d.e. to an identity, is said to be a
solution of the d.e..