1. First-Order Differential Equations
S.-Y. Leu
Sept. 28, 2005

2. CHAPTER 2 First-Order Differential Equations
2.1 Solution Curves Without the Solution
2.2 Separable Variables
2.3 Linear Equations
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Solution
2.7 Linear Models
2.8 Nonlinear Models
2.9 Systems: Linear and Nonlinear Models

3. 2.1 Solution Curves Without the Solutionx y
Slope=
Short tangent segments suggest the shape of the curve 輪廓

4. 2.1 Solution Curves Without the Solution
The general first-order differential equation has the form
F(x, y, y’)=0
or in the explicit form
y’=f(x,y)
Note that, a graph of a solution of a first-order differential equation is called a solution curve or an integral curve of the equation.
On the other hand, the slope of the integral curve through a given point (x0,y0) is y’(x0).

5. 2.1 Solution Curves Without the Solution
A drawing of the plane, with short line segments of slope drawn at selected points , is called a direction field or a slope field of the differential equation .
The name derives from the fact that at each point the line segment gives the direction of the integral curve through that point. The line segments are called lineal elements.

6. 2.1 Solution Curves Without the Solution
Plotting Direction Fields
1st Step
y’=f(x,y)=C=constant curves of equal inclination
2nd Step
Along each curve f(x,y)=C, draw lineal elements
 direction field
3rd Step
Sketch approximate solution curves having the directions of the lineal elements as their tangent directions.

7. 2.1 Solution Curves Without the Solution
If the derivative dy/dx is positive (negative) for all x in an interval I, then the differentiable function y(x) is increasing (decreasing) on I.

8. 2.1 Solution Curves Without the Solution
DEFINITION: autonomous DE
A differential equation in which the independent
variable does not explicitly appear is known
as an autonomous differential equation.
For example, a first order autonomous DE has
the form
DEFINITION: critical point
A critical point of an autonomous DE is a real
number c such that f(c) = 0.
Another name for critical point is stationary point or
equilibrium point.
If c is a critical point of an autonomous DE, then y(x) = c is a
constant solution of the DE.

9. 2.1 Solution Curves Without the Solution
DEFINITION: phase portrait
A one dimensional phase portrait of an
autonomous DE is a diagram which indicates
the values of the dependent variable for which y is
increasing, decreasing or constant.
Sometimes the vertical line of the phase portrait is
called a phase line.

10. DEFINITION: Separable DE
2.2 Separable Variables
DEFINITION: Separable DE
A first-order differential equation of the form
is said to be separable or to
have separable variables.
(Zill, Definition 2.1, page 44).

11. 2.2 Separable Variables
Method of Solution 
If represents a solution

12. 2.2 Separable Variables domain , , The Natural Logarithm
The natural exponential function
domain

13. DEFINITION: Linear Equation
2.3 Linear Equations
DEFINITION: Linear Equation
A first-order differential equation of the form
is said to be a linear equation.
(Zill, Definition 2.2, page 51).
When homogeneous
Otherwise it is non-homogeneous.

14. 2.3 Linear Equations
Standard Form 

15. 2.2 Separable Equations
A differential equation is called separable if it can be written as
Such that we can separate the variables and write
We attempt to integrate this equation

16. 2.2 Separable Equations Example 1. is separable. Write as
Integrate this equation to obtain
or in the explicit form
What about y=0 ? Singular solution !

17. 2.2 Separable Equations Example 2. is separable, too. We write
Integrate the separated equation to obtain
The general solution is
Again, check if y=-1 is a solution or not ?
it is a solution, but not a singular one, since it is a special case of the general solution 

18. 2.3 Linear Differential Equations
A first-order differential equation is linear if it has the form
Multiply the differential equation by to get
Now integrate to obtain
The function is called an integrating factor for the differential equation. 

19. 2.3 Linear Differential Equations
Linear: A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables.
Non-linear: Differential equations that do not satisfy the definition of linear are non-linear.
Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.

20. 2.3 Linear Differential Equations
Example
is a linear DE. P(x)=1 and q(x)=sin(x), both continuous for all x.
An integrating factor is
Multiply the DE by to get Or
Integrate to get
The general solution is

21. 2.3 Linear Differential Equations
Example
Solve the initial value problem
It can be written in linear form
An integrating factor is for
Multiply the DE by to get Or
Integrate to get ,then
for
For the initial condition, we need
C=17/4 the solution of the initial value problem is