MAT 1235Calculus II

Section 9.3

Separable Equations I

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HW and …

WebAssign 9.3 Part I(15 problems, 123min.)

Wednesday: 9.5 Thursday: 9.3 Part II

Differential Equations

Let be a function in A D.E. is an equation involves

, , , , … Our goal is to solve for the solutions

Differential Equations

Many “phenomenon” can be modeled /described by D.E.

Example 0.1 (Chemistry)

Reaction

DCBA = amount of

The rate of formation of is given by

))(( XXkdt

dX

?X t

Example 0.2

ghA

A

dt

dh

w

h 2

?h t

Example 0.3 Pendulum

l0

2

2

l

g

dt

d

?t

Example 0.4

)(1

2

2

tEqCdt

dqR

dt

qdL

q = charge on the capacitor

?q t

Toolbox Approach

Given a differential equation Identify the type/nature of the differential

equation. Use the specified techniques to solve for

the solutions.

Separable Equations

)()( yhxgdx

dy

Technique: Separation of Variables

Example 1

6dy

xydx

1. is called the general solutions of the D.E.

2. We can verify the solution by differentiation.

Remarks23xCey

xyy 6

3. The value of C can be fixed if additional condition is given.

e.g. y(0)=4 (initial condition)

Remarks xyy 6

23xCey

3. The value of C can be fixed if additional condition is given.

e.g. y(0)=4 (initial condition)

4. is called the particular solution of the D.E.

Remarks xyy 6

234 xey

Solution Curves

Condition Initial

234 xy e23xCey

Example 2

42 ydx

dy

2

2

1

4 2 2

2 2

4

1 2 2

A B

y y y

A y B y

y

A y B y

2

1

4 2 2y y y

Remarks 4

24

2 14,

1

x

x

dy Cey y

dx Ce

2 are solutions of the D.E.

which cannot be obtain from the solution above.

2 are called

Why?(Bonus points

t e

)

h

y

y singular solutions.

Example 2

42 ydx

dy 1C 0.1C

0.001C

1C 0.1C

0.001C

Example 3

)(

cos2)2( is

2sin)cos

D.E. theof solutions general The

SolutionsImplicit

Cxye

xdx

dyyex(e

y

yy

Example 3

cos ) sin

(2 ) 2c

2

os

y y

ye

dyx(e ye x

d

y x C

x