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Lecture (25): Ordinary Differential Equations (1 of 2)
A differential equation is an algebraic equation that contains some derivatives:
037352
2
ydx
dy
dx
ydty
dt
dy
• Recall that a derivative indicates a change in a dependent variable with respect to an independent variable.
• In these two examples, y is the dependent variable and t and x are the independent variables, respectively.
Why study differential equations?
• Many descriptions of natural phenomena are relationships (equations) involving the rates at which things happen (derivatives).
• Equations containing derivatives are called differential equations.
• Ergo, to investigate problems in many fields of science and technology, we need to know something about differential equations.
Why study differential equations?
• Some examples of fields using differentialequations in their analysis include:
— Solid mechanics & motion
— heat transfer & energy balances
— vibrational dynamics & seismology
— aerodynamics & fluid dynamics
— electronics & circuit design
— population dynamics & biological systems
— climatology and environmental analysis
— options trading & economics
Examples of Fields Using Differential Equations in Their Analysis
CoolBath
HotBath
kydt
dyckx
dt
dxc
dt
xdm
2
2
02
2
kxdt
dxc
dt
xdm
ikkxdt
dxc
dt
xdm f
2
2
ghACdt
dhA od 2
70 Hkdt
dH
Differential Equation Basics
• The order of the highest derivative ina differential equation indicates the order of the equation.
D.E. PartialOrder Second 01
EquationOrder Second 037
EquationOrder First 35
2
2
2
2
t
T
x
T
ydx
dy
dx
yd
tydt
dy
Simple Differential Equations
A simple differential equation has the form
( )dy
f xdx
Its general solution is
( )y f x dx
Ex. Find the general solution to
Simple Differential Equations
dxxxdy 32 42
32 42 xxdx
dy
dxxxy 32 42
Cxxy 43
3
2
Ex. Find the general solution to
Simple Differential Equations
dxxx
dy
2
1
xxdx
dy2
1
dxx
xy 2
1
Cxxy 2ln
Find the general solution to
Exercise: (Waner, Problem #1, Section 7.6)
dxxxdy 2
xxdx
dy 2
dxxxdxxxy 2122 /
Cxxy 233
3
2
3
1 /
A drag racer accelerates from a stop so that its speed is 40t feet per second t seconds after starting. How far will the car go in 8 seconds?
Example: Motion
seconds.in timeis and
feet,in distance theis )( wher , 40
t
tstdt
ds
ft ?8 s
Given:
Find:
dttds 40
tdt
ds40
Ctdttts 22040
ft 12808208 2 s
Solution:
Apply the initial condition: s(0) = 0
Cs 202000 0C
220tts
The car travels 1280 feet in 8 seconds
Find the particular solution to
dxxxdy 23
0 when 1 ;23 xyxxdx
dy
Cxxdxxxy 243
4
12
14
1 24 xxy
Exercise: (Waner, Problem #11, Section 7.6)
Apply the initial condition: y(0) = 1
C 24 004
11 1C
Separable Differential Equations
A separable differential equation has the form ( ) ( )dy
f x g ydx
Its general solution is1
( )( )dy f x dx
g y
Consider the differential equation 2y
x
dx
dy
Example: Separable Differential Equation
a. Find the general solution.
b. Find the particular solution that satisfies the initial condition y(0) = 2.
dxxdyy 2
Solution:
Step 1 — Separate the variables:
dxxdyy2Step 2 — Integrate both sides:
Step 3 — Solve for the dependent variable:
Cxy
23
23
a.
312
312
2
33
2
3//
DxCxy
This is the general solution
Solution: (continued)
Apply the initial (or boundary) condition, that is, substituting 0 for x and 2 for y into the general solution in this case, we get
3131
202
32 /
/
DD
Thus, the particular solution we are looking for is
823 D
b.
312 8
2
3/
xy
Find the general solution to
Exercise: (Waner, Problem #4, Section 7.6)
x
dx
y
dy
x
y
dx
dy
x
dx
y
dy
) (where CeAAxy
Cxy lnln
xeey CCx ln
Axxey C