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Chapter 1: First-Order Differential Equations

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Chapter 1: First-Order Differential Equations

Sec 1.4: Separable Equations and Applications

Definition 2.1

A 1st order De of the form

is said to be separable.

1

2

3

3

Sec 1.2

How to Solve ?

Sec 1.4: Separable Equations and Applications

1

2

3

4

Solve the differential equation

It may or may not possible to express y in terms of x (Implicit Solution)

Sec 1.4: Separable Equations and Applications

Solve the IVP

Implicit Solutions and Singular Solutions

Solve the IVP

Implicit So , Particular, sol

2

-2

2

-2

Sec 1.2

How to Solve ?

Remember division

3) Remember division

Implicit Solutions and Singular Solutions

Singular Sol

division

Solve the IVP

a general Sol

Family of sol (c1,c2,..)

a general Sol

Family of sol (c1,c2,..)

Particular Sol

No C

The general Sol

It is a general sol

Contains every particular sol

Singular Sol

no value of C gives this sol

Sec 1.4: Separable Equations and Applications

1

2

3

4

Solve the differential equation

It may or may not possible to express y in terms of x (Implicit Solution)

Modeling and Separable DE

Cooling and Heating

Natural Growth and Decay

According to Newton’s Law of cooling

The Differential Equation

K a constant

- serves as a mathematical model for a remarkably wide range of natural phenomena.
- Population Growth
- Compound Interest
- Radioactive Decay
- Drug Elimination

Torricelli’s Law

Water tank with hole

The population of a town grows at a rate proportional to the population present at time t. the initial population of 500 increases by 15% in 10 years. What will be the population in 40 years?

The Differential Equation

K a constant

The Differential Equation

K a constant