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Introduction Introduction
Exact Differential Equations Exact Differential Equations
Bernoulli’s Differential Equation Bernoulli’s Differential Equation
Outline
Math 337 - Elementary Differential Equations
Lecture Notes – Exact and Bernoulli Differential Equations
1 Introduction
Joseph M. Mahaffy, 2 Exact Differential Equations
Gravity
hjmahaffy@sdsu.edui Potential Function
Exact Differential Equation
Department of Mathematics and Statistics
Dynamical Systems Group
Computational Sciences Research Center 3 Bernoulli’s Differential Equation
San Diego State University Logistic Growth Equation
San Diego, CA 92182-7720 Alternate Solution
http://jmahaffy.sdsu.edu Bernoulli’s Equation
Spring 2022
Lecture Notes – Exact and Bernoulli Differential Equations Lecture Notes – Exact and Bernoulli Differential Equations
Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(1/26) Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(2/26)
Introduction Introduction Gravity
Exact Differential Equations Exact Differential Equations Potential Function
Bernoulli’s Differential Equation Bernoulli’s Differential Equation Exact Differential Equation
Introduction Exact Differential Equations
Exact Differential Equations - Potential functions
Introduction
In physics, conservative forces lead to potential functions,
Exact Differential Equations where no work is performed on a closed path
Potential Functions Alternately, the work is independent of the path
Gravity
Bernoulli’s Differential Equation Potential functions arise as solutions of Laplace’s equation in
Applications PDEs
Logistic Growth Potential function are analytic functions in Complex Variables
Naturally arise from implicit differentiation
Lecture Notes – Exact and Bernoulli Differential Equations Lecture Notes – Exact and Bernoulli Differential Equations
Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(3/26) Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(4/26)
Introduction Gravity Introduction Gravity
Exact Differential Equations Potential Function Exact Differential Equations Potential Function
Bernoulli’s Differential Equation Exact Differential Equation Bernoulli’s Differential Equation Exact Differential Equation
Gravity Gravity
Gravity
The force of gravity between two objects mass m and m satisfy
1 2 Differential Equation for Gravity
F(x,y) = Gm m xi + yj
1 2 2 2 3/2 2 2 3/2 The differential equation for gravity is
(x +y ) (x +y )
The potential energy satisfies Gm m x + y dy=0
1 2 2 2 3/2 2 2 3/2 dx
Gm m (x +y ) (x +y )
U(x,y) = − 1 2
2 2 1/2
(x +y ) By the way this problem was set up, the solution is the implicit
Perform Implicit differentiation on U(x,y), where we let y potential function
depend on x (path y(x) depends on x): Gm m
U(x,y(x)) = − 1 2 =C
2 2 1/2
dU(x,y) x y dy (x +y (x))
=Gmm +
dx 1 2 2 2 3/2 2 2 3/2 dx
(x +y ) (x +y )
Aconservative function satisfies dU = 0
dx
Lecture Notes – Exact and Bernoulli Differential Equations Lecture Notes – Exact and Bernoulli Differential Equations
Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(5/26) Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(6/26)
Introduction Gravity Introduction Gravity
Exact Differential Equations Potential Function Exact Differential Equations Potential Function
Bernoulli’s Differential Equation Exact Differential Equation Bernoulli’s Differential Equation Exact Differential Equation
Gravity Gravity
Potential Function Definition
Consider a potential function, φ(x,y) Suppose there is a function φ(x,y) with
By implicit differentiation ∂φ ∂φ
∂x =M(x,y) and ∂y =N(x,y).
dφ(x,y) = ∂φ + ∂φ dy
dx ∂x ∂y dx The first-order differential equation given by
If the potential function satisfies φ(x,y) = C (level potential M(x,y)+N(x,y)dy =0
field), then dx
dφ(x,y) = 0
dx is an exact differential equation with the implicit solution
satisfying:
This gives rise to an Exact differential equation φ(x,y) = C.
Lecture Notes – Exact and Bernoulli Differential Equations Lecture Notes – Exact and Bernoulli Differential Equations
Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(7/26) Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(8/26)
Introduction Gravity Introduction Gravity
Exact Differential Equations Potential Function Exact Differential Equations Potential Function
Bernoulli’s Differential Equation Exact Differential Equation Bernoulli’s Differential Equation Exact Differential Equation
Example 1 Example 2
Example (cont): Begin with
Example: Consider the differential equation: ∂φ =M(x,y)=2x+ycos(xy).
∂x
(2x+ycos(xy))+(4y+xcos(xy)) dy = 0 Integrate this with respect to x, so
dx Z 2
φ(x,y) = (2x+ycos(xy))dx = x +sin(xy)+h(y),
This equation is clearly nonlinear and not separable.
Wehopethat it might be exact! where h(y) is some function depending only on y
If it is exact, then there must be a potential function, φ(x,y) Similarly, we want
satisfying: ∂φ
∂y =N(x,y)=4y+xcos(xy).
∂φ =2x+ycos(xy) and ∂φ =4y+xcos(xy).
∂x ∂y Integrate this with respect to y, so
φ(x,y) = Z (4y +xcos(xy))dy = 2y2 +sin(xy)+k(x),
where k(x) is some function depending only on x
Lecture Notes – Exact and Bernoulli Differential Equations Lecture Notes – Exact and Bernoulli Differential Equations
Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(9/26) Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(10/26)
Introduction Gravity Introduction Gravity
Exact Differential Equations Potential Function Exact Differential Equations Potential Function
Bernoulli’s Differential Equation Exact Differential Equation Bernoulli’s Differential Equation Exact Differential Equation
Example 3 Potential Example
Graph of the Potential Function
Example (cont): The potential function, φ(x,y) satisfies
Potential
2 2 70
φ(x,y) = x +sin(xy)+h(y) and φ(x,y) = 2y +sin(xy)+k(x)
for some h(y) and k(x) 80 60
70
Combining these results yields the solution 60 50
50
al
2 2 i
t
φ(x,y) = x +2y +sin(xy) = C. n40 40
e
ot
P30
Implicit differentiation yields: 20 30
10
dφ dy 0 20
dx =(2x+ycos(xy))+(4y+xcos(xy)) dx =0, 4
2 4 10
0 2
the original differential equation. −2 0
−2 0
y −4 −4 x
Lecture Notes – Exact and Bernoulli Differential Equations Lecture Notes – Exact and Bernoulli Differential Equations
Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(11/26) Joseph M. Mahaffy, hjmahaffy@sdsu.edui —(12/26)
Introduction Gravity Introduction Gravity
Exact Differential Equations Potential Function Exact Differential Equations Potential Function
Bernoulli’s Differential Equation Exact Differential Equation Bernoulli’s Differential Equation Exact Differential Equation
Potential Example Exact Differential Equation
Contour of the Potential Function
Potential Theorem
5 70 Let the functions M, N, M , and N (subscripts denote partial
y x
4 derivatives) be continuous in a rectangular region
3 60 R:α
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