Differential Equations: Basic Definitions and Classifications
A differential equation is an equation involving some function of interest along with a few of its derivatives. Typically, the
function is unknown, and the challenge is to determine what that function could possibly be.
Differential equations can be classified either as “ordinary” or as “partial”.
Ordinary differential equation - A differential equation involving only ordinary derivatives with respect to a single
independent variable
If an equation involves the derivative of one variable with respect to another, then the former is called a
dependent variable and the latter an independent variable
𝑑2𝑥
𝑑𝑥
+ 𝑎 𝑑𝑡 + 𝑘𝑥 = 0
𝑑𝑡 2
Example 1:
Dependent Variable
Independent Variable
𝜕𝑢
𝜕𝑢
− 𝜕𝑦 = 𝑥 − 2𝑦
𝜕𝑥
Example 2:
Dependent Variable
Independent Variable
Partial differential equation - A differential equation involving partial derivatives with respect to more than one
independent variable
Notations
Leibniz:
𝑑𝑦 𝑑 2 𝑦 𝑑 3 𝑦
,
,
,…
𝑑𝑥 𝑑𝑥 2 𝑑𝑥 3
Prime:
𝑦 ′ , 𝑦 ′′ , 𝑦 ′′′ , 𝑦 (4) , 𝑦 (5) , … , 𝑦 (𝑛)
Newton’s dot notation (the “flyspeck” notation):
𝑠,̇ 𝑠̈ , 𝑥̇ , 𝑥̈
Determine the independent and dependent variable and state whether the equation is ordinary or partial.
1.
𝑑𝑦
= 4𝑥 3
𝑑𝑥
2. 𝑦 ′ + 𝑃(𝑥)𝑦 = 𝑄(𝑥)
3.
𝜕2 𝑤
𝜕2 𝑤
= 𝑎 2 𝜕2 𝑥
𝜕𝑡 2
4.
𝑑4𝑦
= 𝑤(𝑥)
𝑑𝑥 4
5. 𝑦 ′′′ − 3𝑦 ′ + 2𝑦 = 0
6.
𝜕2 𝑢
𝜕2 𝑢
𝜕2 𝑢
+ 2+ 2 =0
2
𝜕𝑥
𝜕𝑦
𝜕𝑧
7.
𝑑𝑢
𝑑𝑣
+ = 4𝑢 + 𝑣
𝑑𝑡
𝑑𝑡
8.
𝜕2 𝑢
𝜕2 𝑢
𝜕𝑢
= 2 −4
2
𝜕𝑥
𝜕𝑡
𝜕𝑡
9.
𝜕𝑢
𝜕𝑣
= − 𝜕𝑥
𝜕𝑦
10. 𝑑𝑥 2 − 𝑑𝑥 + 6𝑦 = 0
𝑑2𝑦
𝑑𝑦
Differential equations are also classified by their “order”.
CLASSIFICATION BY ORDER
The order of a differential equation is simply the order of the highest order derivative
explicitly appearing in the equation
𝑑𝑦
= 8𝑥 2
𝑑𝑥
First-Order:
,
𝑑𝑦 3
+ 𝑦 = 𝑥3
𝑑𝑥 𝑥
Second-Order:
𝑑2 𝑦
𝑑𝑦
𝑑2 𝑦
𝑑𝑦 4
−3
+ 4𝑦 = 10 cos(2𝑥) ,
− 2 ( ) − 2𝑦 = 3𝑒 4𝑥
𝑑𝑥 2
𝑑𝑥
𝑑𝑥 2
𝑑𝑥
Third-Order:
𝑑3 𝑦
= 1 + 𝑒 2𝑥
𝑑𝑥 3
𝑑3 𝑦
𝑑2 𝑦 𝑑𝑦
−
2
+
+ 𝑦 = . 8𝑥 2
𝑑𝑥 3
𝑑𝑥 2 𝑑𝑥
,
Exercise 1: What is the order of each of the following equations?
𝑑2𝑦
𝑑𝑦
1. 𝑑𝑥 2 + 7 (𝑑𝑥 )2 + 8𝑦 = sin 2𝑥
𝑑5𝑦
Order: ______________
𝑑3𝑦
2. 𝑑𝑥 5 − cos(𝑥) 𝑑𝑥 3 = 𝑦 2
𝑑 10 𝑥
Order: ______________
𝑑2𝑥
3. ( 𝑑𝑡 10 )5 = ( 𝑑𝑡 2 )7
Order: ______________
CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation is said to be linear if F is linear in
𝑦, 𝑦 ′ , 𝑦 ′′ , … 𝑦 (𝑛) . A linear differential equation is one in which the dependent variable 𝑦 and its derivatives appear in
additive combinations of their first powers.
𝒂𝒏 (𝒙)
𝒅𝒏 𝒚
𝒅𝒏−𝟏 𝒚
𝒅𝒚
(𝒙)
+
𝒂
+ ⋯ + 𝒂𝟏 (𝒙)
+ 𝒂𝟎 (𝒙)𝒚 = 𝑭(𝒙)
𝒏−𝟏
𝒏
−𝟏
𝒅𝒙
𝒅𝒙
𝒅𝒙
Where 𝒂𝒏 (𝒙), 𝒂𝒏−𝟏 (𝒙), … , 𝒂𝟎 (𝒙) and 𝐹(𝑥) depend only on the independent variable 𝑥. The additive combinations are
permitted to have multipliers (coefficients) that depend on x; no restrictions are made on the nature of this 𝑥dependence. If an ordinary differential equation is not linear, then we call it nonlinear.
The two properties of a linear ODE are as follows:
•
•
The dependent variable 𝑦 and all its derivatives 𝑦, 𝑦′ , . . . , 𝑦 (𝑛) are of the first degree, that is, the power
of each term involving 𝑦 is 1
The coefficients 𝑎0 , 𝑎1 , … , 𝑎𝑛 of 𝑦 , 𝑦 ′ , … , 𝑦 (𝑛) depend at most on the independent variable 𝒙.
Linear
(𝑦 − 𝑥)𝑑𝑥 + 2𝑥𝑑𝑦 = 0 ,
Non-Linear
𝑑4 𝑦
− 𝑦3 = 0
𝑑𝑥 4
,
𝑑3 𝑦
𝑑𝑦
+
𝑥
+ 6𝑦 = 𝑒 3𝑥
𝑑𝑥 3
𝑑𝑥
(1 − 𝑦)𝑦 ′ + 2𝑦 = 𝑒 𝑥
State whether the equation is linear or non-linear.
𝑑𝑥
1. 𝑡 3 𝑑𝑡 = 𝑡 3 + 𝑥
2. (𝑦 − 2𝑥)𝑑𝑥 − 3𝑥𝑑𝑦 = 0
3. 𝑦 ′′′ + 4𝑦 ′′ − 10𝑦 ′ + 7 = 0
4.
5. 𝑦𝑦 ′′ = 𝑥
6. 𝑦 ′′ + 2𝑦 ′ − 8𝑦 = 𝑥 2 + 𝑐𝑜𝑠 𝑥
7.
𝑑𝑦
= 1 − 𝑥𝑦 + 𝑦 2
𝑑𝑥
8.
𝑑𝑤 4
𝑑3𝑤
𝑑2𝑥
+ 𝑘2𝑥 = 0
𝑑𝑡 2
𝑑2𝑦
+ 𝑦3 = 0
𝑑𝑥 2
10. 𝑥(𝑦′′)3 + (𝑦′)4 − 𝑦 = 0
9. ( 𝑑𝑥 3 )2 − 2 ( 𝑑𝑥 ) + 𝑦𝑤 = 0
Solutions
Any function that satisfies a given differential equation is called a solution to that differential equation.
Example 1. Consider the differential equation
𝑑𝑦
− 3𝑦 = 0
𝑑𝑥
o
o
𝑦(𝑥) = 𝑒 3𝑥 is a solution to the differential equation
𝑦(𝑥) = 𝑥 3 is not a solution to the differential equation
EXPLICIT AND IMPLICIT SOLUTIONS
Explicit Solution.
A solution in which the dependent variable is expressed solely in terms of the independent
variable and constants
DIFFERENTIAL EQUATION
1.
EXPLICIT SOLUTION
𝑑𝑦
= 𝑥𝑦 1/2
𝑑𝑥
𝑦=
1 4
𝑥
16
2. 𝑦 ′′ − 2𝑦 ′ + 𝑦 = 0
𝑦 = 𝑥𝑒 𝑥
3. 𝑥𝑦 ′ + 𝑦 = 0
𝑦 = 1/𝑥
A relation 𝐺(𝑥, 𝑦) = 0 is said to be an implicit solution on the interval if it defines one or more explicit solutions on I.
Example :
Show that 𝑥 + 𝑦 + 𝑒 𝑥𝑦 = 0 is an implicit solution to the nonlinear equation
(1 + 𝑥𝑒 𝑥𝑦 )
𝑑𝑦
+ 1 + 𝑦𝑒 𝑥𝑦 = 0
𝑑𝑥
Typically, a differential equation will have many different solutions. Any formula (or set of formulas) that describes all
possible solutions is called a general solution to the equation.
Particular Solution.
A solution of a differential equation that is free of arbitrary parameters
Initial-Value Problems
A solution to an initial-value problem is a solution to the differential equation that also satisfies the given initial values.
The usual approach to solving such a problem is to first find the general solution to the differential equation (via any of
the methods we’ll develop later), and then determine the values of the arbitrary constants in the general solution so that
the resulting function also satisfies each of the given initial values.
Example: Consider the initial value problem
𝑑𝑦
= 6𝑥 with 𝑦(1) = 8
𝑑𝑥