Chap9_Sec3 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

Info iconThis preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
DIFFERENTIAL EQUATIONS We have looked at first-order differential equations from a geometric point of view (direction fields) and from a numerical point of view (Euler’s method). What about the symbolic point of view?
Background image of page 2
It would be nice to have an explicit formula for a solution of a differential equation. Unfortunately, that is not always possible. DIFFERENTIAL EQUATIONS
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
9.3 Separable Equations In this section, we will learn about: Certain differential equations that can be solved explicitly. DIFFERENTIAL EQUATIONS
Background image of page 4
A separable equation is a first-order differential equation in which the expression for dy / dx can be factored as a function of x times a function of y . In other words, it can be written in the form ( ) ( ) dy g x f y dx = SEPARABLE EQUATION
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The name separable comes from the fact that the expression on the right side can be “separated” into a function of x and a function of y . SEPARABLE EQUATIONS
Background image of page 6
Equivalently, if f ( y ) ≠ 0, we could write where ( ) ( ) dy g x dx h y = ( ) 1/ ( ) h y f y = SEPARABLE EQUATIONS Equation 1
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
To solve this equation, we rewrite it in the differential form h ( y ) dy = g ( x ) dx so that: All y ’s are on one side of the equation. All x ’s are on the other side. SEPARABLE EQUATIONS
Background image of page 8
Then, we integrate both sides of the equation: ( ) ( ) h y dy g x dx = SEPARABLE EQUATIONS Equation 2
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Equation 2 defines y implicitly as a function of x . In some cases, we may be able to solve for y in terms of x . SEPARABLE EQUATIONS
Background image of page 10
We use the Chain Rule to justify this procedure. If h and g satisfy Equation 2, then ( 29 ( 29 ( ) ( ) d d h y dy g x dx dx dx = SEPARABLE EQUATIONS
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Thus, This gives: Thus, Equation 1 is satisfied. ( 29 ( ) ( ) d dy h y dy g x dy dx = ( ) ( ) dy h y g x dx = SEPARABLE EQUATIONS
Background image of page 12
a.Solve the differential equation b.Find the solution of this equation that satisfies the initial condition y (0) = 2. 2 2 dy x dx y = SEPARABLE EQUATIONS Example 1
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
We write the equation in terms of differentials and integrate both sides: y 2 dy = x 2 dx y 2 dy = x 2 dx y 3 = x 3 + C where C is an arbitrary constant. SEPARABLE EQUATIONS Example 1 a
Background image of page 14
We could have used a constant C 1 on the left side and another constant C 2 on the right side. However, then, we could combine these constants by writing C = C 2 C 1. SEPARABLE EQUATIONS Example 1 a
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
y , we get: We could leave the solution like this or we could write it in the form where K = 3 C . Since
Background image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 61

Chap9_Sec3 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online