University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2020
Views:

66

Pages:

1
Author:

ramramum4x

Summary: Autonomous D.E.s (2.6) 1. Definition: An ODE is said to be autonomous if it can be written as y 0 = f (y). 2. Definition: An equilibrium solution is a constant solution, y(t) = c, where f (c) = 0. That is, you solve for equilibrium solutions by solving f (y) = 0 for y. 3. Definition: Let y(t) = c be an equilibrium solution. Then it is said to be asymptotically stable if all nearby solutions tend towards the equilibrium as t → ∞. df Algebraically, this means dy (c) < 0. The equilibrium y(t) = c is said to be asymptotically unstable if all nearby solutions tend away from y = c as t → ∞. Algedf df braically, dy (c) < 0. NOTE: If dy (c) = 0, these algebraic tests don’t work. Look at the phase diagram to get the stability. 4. Remark: Since y 0 = f (y), we can plot f (y) versus y. Such a plot is called a phase diagram. From the phase diagram, it is possible to construct a sketch of y(t) on the direction field. From the phase diagram, we can determine where y(t) is increasing, decreasing and concave up or down. 5. Remark: Since y 00 (t) = df dy dy dt we can determine concavity of y(t) by considering both the local slopes on the phase diagram and the sign of y 0 (t). 6. Remark: If y 0 = f (y), the slopes in the direction field are independent of time. Therefore, on the direction field, the slopes along any horizontal line are identical. Some implications: • One can determine if a direction field is consistent with an autonomous differential equation. • One can quickly sketch the direction field for an autonomous differential equation. • Solutions to autonomous first order differential equations cannot be periodic. 7. Note that all autonomous ODEs are seperable, so if we can find an antiderivative 1 for f (y) , we can solve all autonomous first order differential equations. 1

Differential Equations Summary Autonomous D.E.s (2.6)

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