Chap9_Sec3

# Chap9_Sec3 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9

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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?
It would be nice to have an explicit formula for a solution of a differential equation. Unfortunately, that is not always possible. DIFFERENTIAL EQUATIONS

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9.3 Separable Equations In this section, we will learn about: Certain differential equations that can be solved explicitly. DIFFERENTIAL EQUATIONS
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

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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
Equivalently, if f ( y ) ≠ 0, we could write where ( ) ( ) dy g x dx h y = ( ) 1/ ( ) h y f y = SEPARABLE EQUATIONS Equation 1

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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
Then, we integrate both sides of the equation: ( ) ( ) h y dy g x dx = SEPARABLE EQUATIONS Equation 2

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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
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

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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
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

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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
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

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y , we get: We could leave the solution like this or we could write it in the form where K = 3 C . Since
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## This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap9_Sec3 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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