Download Notes_Chapter_1

Differential Equations
Chapter 1
• A differential equation in x and y is an
equation that involves x, y, and
derivatives of y.
• A mathematical model often takes the
form of a differential equation.
• A function f is called a solution of a
differential equation if the equation is
satisfied when y = f(x) and its derivatives
are substituted into the equation.
Example
Show that every member of the family of functions
1  ce
y
t
1  ce
t
Is a solution of the differential equation


1 2
y  y  1
2
Newton’s Law of Cooling
• Newton’s Law of Cooling states that the
rate of cooling of an object is proportional
to the temperature difference between the
object and its surroundings, provided that
this difference is not too large.
• Let T(t) be the temperature of the object at
time t and Ts be the temperature of the
surroundings.
dT
 k T  TS 
dt
Where k is a constant.
Orthogonal Trajectories
• An orthogonal trajectory of a family of
curves is a curve that intersects each
curve of the family orthogonally, that is, at
right angles.
• Orthogonal trajectories arise in various
branches of physics. For example, in an
electrostatic field, the lines of force are
orthogonal to the lines of constant
potential.
Example
Find the equation of the orthogonal trajectories to the given
family of curves.
c
y
x
Hooke’s Law
• The restoring force of a spring is directly
proportional to the displacement of the
spring from its equilibrium position and is
directed toward the equilibrium position.
• Let y(t) denote the displacement of the
spring from its equilibrium position at time
t.
Hooke’s Law
FS   ky
2
d y
m 2   ky
dt
• An ordinary differential equation (DE) is
one in which the unknown function y(x)
depends only on one variable, x.
• The order of the highest derivative
occurring in a DE is called the order of the
DE.
• A differential equation that can be written in the
form
a0 x  y n   a1 x  y n 1    an x  y  F ( x)
Where a0,,a1, …, an and F are functions of x only, is called a
linear DE of order n. Such a DE is linear in y and its derivatives.
• A solution to an nth-order DE on an
interval I is called the general solution on I
if it satisfies the following conditions:
– The solution contains n constants.
– All solutions to the DE can be obtained by
assigning appropriate values to the constants.
• A solution to a DE is called a particular
solution if it does not contain any arbitrary
constants not present in the DE itself.
• Unfortunately, it’s impossible to solve most
differential equations in the sense of
obtaining an explicit formula for the
solution.
• Despite the absence of an explicit solution,
we can still learn a lot about the solution
through a graphical approach (directions
fields) or a numerical approach (Euler’s
method)
Example
• Suppose we are asked to sketch the graph of
the solution of the initial value problem
y  x  y
y 0  1
• We don’t know a formula for the solution, so
how can we possibly sketch its graph?
• What does a DE mean?
• The equation y  x  y
tells us that the
y 0  1
slope at any point (x, y) on the graph (called
the solution curve) is equal to the sum of the xand y-coordinates of the point.
• The graph of a solution of a DE is called a
solution curve of the equation. From a
geometric viewpoint, a solution curve of a
DE is a curve in the plane whose tangent
at each point (x, y) has slope m  dy dx
Graphical Solutions
• Through each of a representative
collection of points (x, y) draw a short line
dy
m

segment having slope
dx
• The set of all these line segments is called
a direction field (or slope field).
• Fairly easy to do with CAS.
Isoclines
• An isocline of the DE dy dx  f x, y  is a curve of
the form f(x, y) = c on which the slope is
constant.
• If these isoclines are simple and familiar
curves, we first sketch several of them,
then draw short line segments with the
same slope c at representative points of
each isocline f(x, y) = c.
• Consider the DE
y2  x2
dy
0
dx
If we write this equation in this form
dy
y2
 2
dx
x
dy
y2
 2
dx
x
dy
dx
 2  2
y
x
dy
dx
 2   2
y
x
1
1

C
y
x
x
y  yx  
Cx  1
Existence & Uniqueness of
Solutions
Suppose that the real valued function f(x, y)
is continuous in the xy-plane containing
the point (a, b) in its interior. Then the
dy
initial value problem dx  f x, y  ya   b
has at least one solution on some interval I
containing the point a. If in addition, the
partial derivative f y
is continuous on
that rectangle, then the solution is unique
on some (perhaps smaller) open interval J
containing the point x = a.
dy
x
 2y
dx
Applying the theorem, we see that this DE has a unique solution near any point
where x ≠ 0.
dy
 H  x, y 
dx
• The first-order DE
is called
separable provided that H(x, y) can be
written as the product of a function of x
and a function y.
dy
g x 
 g  x   y  
dx
f y
1
y 
f y