LECTURE #17: WHAT’S A DIFFERENTIAL EQUATION?
Objectives
1)
2)
3)
4)
To be able to distinguish between O.D.E’s and P.D.E.’s, and linear and nonlinear, differential equations.
To understand the definition of the order of a differential equation.
To know how to show that a given function satisfies a specific differential equation.
To begin to appreciate the “guessing strategy” that we initially apply when solving differential equations.
One of the most common types of equations in mathematics is called a differential equation. A differential
equation is simply an equation that contains derivative notation. Behold some examples of regular old equations
along with some equations that are not considered differential equations:
some non-differential equation equations
some differential equation equations
Sermon #7: There are many kinds of differential equations, and in order to solve them, you gotta know, well, what
they are called. Said differently, if you can’t classify a given differential equation, you can’t in general solve it, and
this would be bad. So it’s terminology-tackling time. Be forewarned that there are way more types of differential
equations than what you’re about to see; don’t worry; we’ll get to others eventually…
Ordinary differential equations, or O.D.E.’s, are ordinary because _______________________________
_______________ Partial differential equations, or P.D.E’s, contain ______________________________ (we won’t
focus much time on these guys). The order of a differential equation is ____________________________________
A linear differential equation is an equation of the form ________________________________________ , whereas a
nonlinear differential equation is not of this form. Informally, a differential equation is linear if _______________
________________________________________________________________________________________________
Example 17.1: Let’s classify each of the following differential equations as ordinary, partial, linear or nonlinear,
and give the order of each (and please pay close attention to the various derivative notations).
differential equation
ordinary vs. partial
linear vs. nonlinear
order
arctan( x + y ) + 4 y′′= xy
y ' ' ' + y = y′ln( x + y )
u xx + u yy = 0
y ' '⋅cos( xy ) − y ' ' ' ' =
∂y
x 2 y − ∂x =
dy
dx
y
3
x +1
∂2 y
sin( x )
∂x 2
= 3y −8
Now…There are three types of “differential equations situations” that will be encountered in the material in
this class. Said differently, with a few exceptions there are basically three kinds of homework and exam questions that
you’ll see. After listing them, let’s see some examples of them…
1)
2)
3)
“Show that the following function satisfies the following differential equation.”
“Sketch the solution to a given differential equation and use the picture to estimate specific solutions.”
“Find one or all solutions to the following differential equation.”
Example 17.2: Show that y = sin(2 x ) + 5 x + 7 is a solution to the equation y (iv ) + 5 y′′+ 4 y = 20 x + 28 .
To answer this question correctly, we need to compute the correct derivatives of the given function y , stick
these things into the given differential equation, and then (hopefully!) verify that the differential equation is indeed
satisfied. So let’s get down to business…
computing our derivatives goo
plugging-derivatives-into-our-differential-equation-goo
y = sin( x) + 5 x + 7
y (iv ) + 5 y′′+ 4 y = 20 x + 28 ⇒
y′= _____________ , y′′= ________
y (iii ) = _________ , y (iv ) = ________
This verification process can get more complicated, depending upon how messy the given function y is,
how messy the given differential equation is, and how messy it is to take the necessary derivatives of y . In general,
however, every such type of question follows this solving pattern, and that’s a good thing.
The second kind of differential equation situation involves drawing pictures called slope fields or direction
fields, together with an estimation technique called Euler’s Method. We’ll investigate this kind of situation soon.
The third kind of differential equation situation tends to be the trickiest of the three. This is because no allencompassing approach to “solving” differential equations exists; moreover, the solving processes often involve
integration, which can be challenging. Fret not; here comes a basic kind of situation #3, umm, well, situation.
Example 17.3: Find a solution to the differential equation y ′− 3 y = 0 .
Hmm…This differential equation shows a linear relationship between a function y and its derivative y ′.
How many functions are multiples of their derivatives?
checking one solution “guess” goo
One possibility is y ( x ) = e x , the glorious exponential function. So a
candidate, or a guess, for a solution to y ′− 3 y = 0 is y ( x ) = e x .
Let’s compute y ′, then stick y ′ and y into y ′− 3 y = 0 and see
what happens…
Oops – wrong guess. It appears that we need to introduce some sort of “fudge factor” into our original guess.
Let’s try this and see what happens…
fudge factor not-so-good-guessing goo
the correct guess at a fudge factor goo
y ( x ) = 3e x ⇒
y ( x ) = e3 x ⇒
1x
y ( x) = e 3 ⇒
Example 17.4: Find all solutions to the differential equation y′− 3x 2 = 0 .
Remember that “finding a solution to a differential equation” means finding a function y = y (x) that
satisfies the given differential equation. So…
finding one solution goo
finding all solutions goo
Finding one solution isn’t all that hard – we can
just modify the appearance of the differential
equation and then solve it via integration. But
the set of all solutions – a family of solutions, if
you will – follows from thinking a little bit. And
yes, just as you’ve seen when dealing with
integration in general, the constant C can be any
real number.