1. Math 3120 Differential Equations with Boundary Value Problems
Chapter 2: First-Order Differential Equations
Section 2-1: Solution Curves without a
solution

2. Methods to find solutions of DE
Analytical Methods: Look for explicit formulas to describe the solution of a DE. For example, separable variable, exact equations, etc.
Qualitative Methods: Geometric techniques for representing equations. For example, Direction fields, phase line, phase portraits.
Numerical Methods: Approximates solutions to IVP using calculator or computers. For example, Euler method, Runge-Kutta method, etc.

3. Direction Field (DF) / Slope fields
Represent the motion of an object falling in the atmosphere near sea level by the DE
Look at the qualitative aspects of solution of Eq. (1) without actually solving them.
Taking g = 9.8 m/sec2, m = 10 kg,  = 2 kg/sec,
we obtain
Investigate the behavior of the solutions of Eq. (2) without solving the DE.

4. Example 1: Sketching Direction Field
Using differential equation and table, plot slopes (estimates) on axes below. The resulting graph is called a direction field. (Note that values of v do not depend on t.)

5. Direction Field & Equilibrium Solution
Arrows give tangent lines to solution curves, and indicate where soln is increasing & decreasing (and by how much).
Horizontal solution curves are called equilibrium solutions.
Use the graph below to solve for equilibrium solution, and then determine analytically by setting v' = 0.

6. Note
Visually the DF suggest the appearance or shape of family of solution curves to the DE.
Each line segment is tangent to the graph of the solutions of DE that passes through that point.
Use these line segments as a guide to sketch the graph of the solutions of the DE.
Look at the Long-term behavior of solutions of the DE.
Use Mathematica to generate a DF.

7. Sketch some Solution Curves
When graphing direction fields, be sure to use an appropriate window, in order to display all equilibrium solutions and relevant solution behavior.

8. Equilibrium Solutions
In general, for a differential equation of the form
find equilibrium solutions by setting y' = 0 and solving for y :
Example: Find the equilibrium solutions of the following.

9. Example 2: Graphical Analysis
Discuss solution behavior as t → ∞ for the differential equation below, using the corresponding direction field.

10. Example 3: Graphical Analysis
Discuss how the solutions behave as t → ∞ for the differential equation below, using the corresponding direction field.

11. Example 4: Graphical Analysis for a Nonlinear Equation
Discuss solution behavior and dependence on the initial value y(0) for the differential equation below, using the corresponding direction field.

12. Autonomous First-Order Differential Equations
In this section, we examine equations of the form
called autonomous equations, where the independent variable t does not appear explicitly.
For Example:
A real number c is called a Critical Point (CP) of the function y' = f (y) if it is a zero of f, i.e., f( c) =0. Critical points are also called equilibrium or stationary points.
Many models of physical laws have DE that are autonomous. For instance, Exponential Growth, Logistic Growth, Newton’s Law of Cooling etc.

13. Phase Lines
Autonomous equations have slope field where the lineal elements (small tangent lines) are parallel along horizontal line in the t-y plane, i.e. two points with the same y-coordinate but different t-coordinate have the same lineal elements.
Thus, there is great deal of redundancy in the slope field of autonomous equations. Thus, we need to draw one vertical line containing the information. This line is called the phase line for the autonomous equation.

14. Phase Portrait
Phase Portraits is a phase line marked by a point where the derivative is zero and also indicate the sign of the derivative on the intervals in between.
The phase portraits provide qualitative information about the solutions of the differential equation

15. How to Draw Phase Portraits
For the autonomous equation.
Draw the y-line ( of the t y-plane).
Find the equilibrium points ( numbers given by f(y)=0), and mark them on the line.
Find the intervals of y-values for which f(y) > 0, and draw arrows pointing up these intervals.
Find the intervals of y-values for which f(y) < 0, and draw arrows pointing down these intervals.
Example: Draw the phase portrait of

16. Attractors and Repellers
Let y(t) be a non-constant solution of the autonomous DE given by y' = f(y) and c be the critical point of the DE.
If all solutions y(t) of (3) that start from an initial point (x0,y0) sufficiently near c exhibit the asymptotic behavior limx→∞ y(t) =c, then c is said to be asymptotically stable. Asymptotically stable critical points are also called attractors .
If all solutions y(t) of (3) that start from an initial point (x0,y0) move away from c as t increases, then c is said to be unstable . Unstable critical points are also called repellers.
If c exhibits characteristic of both an attractor and a repeller, then say that the critical point c is semi-stable.

17. Example 5: CP, Phase Portraits
Find the critical points, phase portrait of the given autonomous DE. Classify each critical points as asymptotically stable, unstable, or semi-stable.

18. How to use Phase Portraits to sketch Solutions
Consider the Differential Equations

19. Summarize
Let y(t) be a non-constant solution of the autonomous DE given by y' = f(y) on a Region R corresponding to an interval I and c1, c2 be the critical points of the DE such that c1< c2 . Then the graph of the equilibrium solutions partition the region r into R1, R2, R3. The following conclusions can be drawn:
y(t) is a solution that passes through (t0,y0) in a sub region Ri ,then y(t) remains in that region for all t.
f(y) cannot change signs in a sub region.
Since y' is either positive or negative in Ri , therefore y(t) is strictly monotonic.
If y(t) is bounded above by a critical point ci , then the graph of y(t) must approach the graph of the equilibrium solutions either as t→∞ or t→-∞ .
If y(t) is bounded below by a critical point ci , then the graph of y(t) must approach the graph of the equilibrium solutions either as t→∞ or t→-∞ .
If y(t) is bounded above and below by two consecutive critical points ci , then the graph of y(t) must approach the graph of both the equilibrium solutions , one as t→∞ and the other as t→-∞ .

20. Example 6: Phase Portraits, Solution Curves
Sketch the phase portrait and solution curves for the following differential equation.

21. Example 7: Qualitative Analysis for the Logistic Population Model
Logistic Population Model states that the rate of growth of the population is proportional to the size of the population with the assumptions that there is limitation of space and resources.
Variables: time t, population P
Parameter: growth rate coefficient k, carrying capacity N.
The Logistic Population Model is given by

22. Warning: Not all Solutions exist for all time
Sketch the phase portrait and solution curves for the following differential equation.
Note: We cannot tell if solutions will blow up in finite time by simply looking at the phase line.