MATH 312 Section 2.4: Exact Differential Equations

Transcription

1 MATH 312 Section 2.4: Exact Differential Equations Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007

2 Outline 1 Exact Differential Equations 2 Solving an Exact DE 3 Making a DE Exact 4 Conclusion

3 A Motivating Our tools so far allow us to solve first-order differential equations which are separable and/or linear. Is the following differential equation separable or linear? (tan x sin x sin y)dx + (cos x cos y)dy = 0 After rewriting as shown, what do you notice? dy dx The equation is not separable. The equation is not linear. = sin x sin y tan x cos x cos y We need a new solution method for this DE!

4 Working Backwards We develop our method using Calculus notation. Differentials Recall that if f (x, y) has continuous first partials on some region of the xy-plane, then with z = f (x, y) the differential is: dz = f f dx + x y dy Why is this of use? Recall our motivating example. Now, to solve (tan x sin x sin y)dx + (cos x cos y)dy = 0 we find an f (x, y) for which f x = (tan x sin x sin y) and = (cos x cos y), and set f (x, y) = c for any constant c so that dz = 0. f y

5 Exact Differentials and Equations We now formalize this type of solution with several definitions. Definition 2.3 A differential expression of the form M(x, y) dx + N(x, y) dy is an exact differential in a region R of the xy-plane if it corresponds to the differential of some function f (x, y) defined on R. Definition 2.3, Part II A first order differential equation of the form M(x, y) dx + N(x, y) dy = 0 is an exact equation if the left side is an exact differential. Solving an Exact Equation If the differential of f (x, y) is M(x, y) dx + N(x, y) dy, then f (x, y) = c is an implicit solution to the DE M(x, y) dx + N(x, y) dy = 0

6 Verifying Exactness We now consider how to tell if a DE is exact. Is the differential equation below exact? Theorem 2.1 (2x 1) dx + (3y + 7) dy = 0 Let M(x, y) and N(x, y) be continuous with continuous first partial derivatives on a rectangular region R of the xy-plane. Then, a necessary and sufficient condition that M(x, y) dx + N(x, y) dy be an exact differential is that M y = N x. Step 1: M y = N x implies exactness. Step 2: exactness implies M y = N x.

7 Solution Method To solve an exact differential equation, we will follow the procedure from Step 2 of the theorem proof. Solution Method To solve an exact differential equation, follow these steps. f x f (x, y) = = M(x, y) M(x, y) dx + g(y) f y = ( y ) M(x, y) dx + g (y) = N(x, y)

8 s Use this procedure to find an implicit solution to each of the following exact differential equations. Solve (2xy 2 3) dx + (2x 2 y + 4) dy = 0 x 2 y 2 3x + 4y = c Solve (tan x sin x sin y) dy + (cos x cos y) dy = 0 cos x sin y ln cos x = c

9 Not Everything is Exact Unfortunately, not every differential equation which has the form M(x, y) dx + N(x, y) dy = 0 is exact. Show that the differential equation below is not exact. (2y 2 + 3x) dx + 2xy dy = 0 Sometimes we can find an integrating factor µ(x, y) so that the equation obtained by multiplying by µ(x, y) (shown below) is exact. µ(x, y)m(x, y) dx + µ(x, y)n(x, y) dy = 0

10 Using an Integrating Factor In order for our integrating factor to work, we need the following to be exact. µ(x, y)m(x, y) dx + µ(x, y)n(x, y) dy = 0 Applying the exactness theorem, this means: y (µm) = x (µn) µ y M + M y µ = µ x N + N x µ µ x N µ ( M y M = µ y N ) x

11 Finding an Integrating Factor To continue, we must assume that µ depends only on x or y. Suppose µ = µ(x). µ x N µ ( M y M = µ y N x ) µ x N = µ dµ dx = ( M y N x ( My N x N ) µ ) Finding µ My Nx Nx My If N depends only on x or if M depends only on y, then we can solve the separable DE dµ My Nx dx = N µ or dµ Nx My dy = M µ respectively to find the integrating factor µ.

12 Completing our Recall our problem example from a few slides previous. Solve (2y 2 + 3x) dx + 2xy dy = 0 M y N x N = 4y 2y 2xy and N x M y M = 2y 4y 2y 2 + 3x dµ dx 1 R µ = 0 is linear, so µ = 1 e x x x(2y 2 + 3x) dx + x(2xy) dy = 0 is exact dx = x x 2 y 2 + x 3 = c

13 Important Concepts Things to Remember from Section Identifying exact differential equations 2 Solution method for exact differential equations 3 Using an integrating factor to make a DE exact