2.2 Separable Equations

Transcription

1 2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve 1. dy dx = (1 + y2 ) cos x 1

2 2. dx dt = x2 Separation of Variables Separate the variables to get Integrate both sides: Solve for y if possible. Example Solve 3. x dy dx = 4y 2

3 Solve the initial value problem. 4. dy dt = y2 1 + t, y(0) = 1 3

4 5. dx dt = 4(x2 + 1), x ( π 4 ) = 1 4

5 6. dy dx = y2 4 5

6 2.3 Linear Equations Definition A first-order differential equation of the form is said to be a linear equation in the dependent variable y. Note: a 1 (x), a 0 (x), and g(x) depend only on the independent variable x in a linear equation A DE where g(x) = 0 is often referred to as homogeneous. Standard Form If we divide everything by a 1 (x) then we get the standard form, written as Examples Solve 1. dy dt + 2ty = 0 6

7 2. dy + y sin x = 0 dx 3. (Optional) dy dt t y = 0 7

8 Using an Integrating Factor to Solve a Linear DE 1. Put the linear equation in standard form to identify P(x). 2. Find the integrating factor 3. Multiply the standard form by the integrating factor. 4. Integrate both sides of the resulting equation. Note: Using this method will give us the general solution, meaning we will have found every solution to the linear DE. Examples Find the general solution using an integrating factor. 4. dy dx + 3y = 2e x 8

9 5. 2y y = 4 sin 3t 9

10 6. x dy dx + 4y = x3 x 10

11 Solve the initial value problem 7. dx dt + 3x = 8et, x(0) = 0 11

12 8. (optional) dy + y tan x = sin x dx 12

13 2.4 Exact Equations 1. Solve the differential equation Suppose there is a function 2xy 9x 2 + (2y + x 2 + 1) dy dx = 0 Don t worry about how we got it for the moment! Then F x and F y are Looking at the original DE you see that F x and F y show up and we can rewrite the equation as From Calculus III recall Thus Exact Equation The differential equation is exact in a rectangle R if there is a function F(x, y) such that For all (x, y) in R. 13

14 Test for Exactness If the first partial derivatives of M(x, y) and N(x, y) are continuous in a rectangle R, then Is an exact equation in R if and only if At every point in R. Thus, if Then there is a function F(x, y) such that And the solution to the differential equation is Examples Solve 1. 2xy 9x 2 + (2y + x 2 + 1) dy dx = 0 14

15 Solving Exact Equations If Mdx + Ndy = 0 is exact, then F x = M. a) Integrate F x = M with respect to x to get b) To determine g(y), take the partial derivative of the above with respect to y to get F y. c) Set equal to F y = N and solve for g (y). d) Integrate g (y) to get g(y) + C. e) Plug g(y) + C into F(x, y) f) The solution to Mdx + Ndy = 0 is 15

16 Examples Determine whether the equation is exact then solve. 2. 2xy + (1 + x 2 ) dy dx = 0 16

17 3. (2xy 2 3)dx + (2x 2 y + 4)dy = 0 17

18 4. Solve the initial value problem. (x + y) 2 dx + (2xy + x 2 1)dy = 0, y(1) = 1 18

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20 2.5 Special Integrating Factors Integrating Factor for Nonexact Equations If the equation Is not exact, but the equation Which results from multiplying by μ(x, y), is exact, then μ(x, y) is called an integrating factor of the equation. Note: Once we multiply by μ(x, y) the test for exactness must be adjusted to Example 1. Verify that the equation is not exact then use the integrating factor μ(x, y) = 1 x2 to make it exact. y x dy dx = 0 To find μ(x, y) in general, we can use the product rule Solving this for μ(x, y) can end up being more work than solving the original DE unless μ(x, y) depends on only one variable. 20

21 Special Integrating Factors M(x, y)dx + N(x, y)dy = 0 a) If M y N x N is continuous and only depends on x, then Is an integrating factor. b) If N x M y M is continuous and only depends on y, then Is an integrating factor. Example - Solve 2. y x dy dx = 0 21

22 3. 6xydx + (4y + 9x 2 )dy = 0 22

23 2.6 Substitutions and Transformations Substitution Procedure a) Identify the type of equation and the appropriate substitution or transformation. b) Rewrite the original equation in terms of new variables. c) Solve the transformed equation. d) Express the solution in terms of the original variables. Definition f is a homogeneous function of degree α iff(tx, ty) = t α f(x, y) for some real number α. Example 1. f(x, y) = x 3 + y 3 Definition A first-order DE is homogeneous if M and N are homogeneous equations of the same degree. So, if If the differential equation is homogeneous then we will use the substitution where u and v are new dependent variables. This will reduce a homogeneous equation to a separable ODE. 23

24 Examples 2. x 2 dy dx = xy + 3y2 on x > 0 24

25 3. xy 2 dy dx = y3 + x 3 x > 0 25

26 Bernoulli Equation A first-order equation that can be written in the form Where P(x) and Q(x) are continuous on an interval (a, b) and n is any real number, is called a Bernoulli equation. Note: the equation is linear if n = 0 or 1. For other cases, the substitution u = y 1 n reduces the equation to a linear equation. Example 4. dy dx 3 + 6y = 30y2 26

27 5. (x 2 + 2y 2 ) dy = xy y( 1) = 1 dx 27

28 Reduction To Separation of Variables A DE of the form Can be reduced to a separable equation using the substitution u = Ax + By + C, B 0 Note: This will be left as an exercise in the homework. 28