Chapter 1: First-Order Differential Equations

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Chapter 1 First-Order Differential Equations
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Sec 1.4 Separable Equations and Applications
Definition 2.1
A 1st order De of the form
is said to be separable.
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Sec 1.2
How to Solve ?
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Sec 1.4 Separable Equations and Applications
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Solve the differential equation
It may or may not possible to express y in terms
of x (Implicit Solution)
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Sec 1.4 Separable Equations and Applications
Solve the IVP
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Implicit Solutions and Singular Solutions
Solve the IVP
Implicit So , Particular, sol
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Sec 1.2
How to Solve ?
Remember division
3) Remember division
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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

1. It is a general sol
2. Contains every particular sol

Singular Sol
no value of C gives this sol
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Sec 1.4 Separable Equations and Applications
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Solve the differential equation
It may or may not possible to express y in terms
of x (Implicit Solution)
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(No Transcript)
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Modeling and Separable DE
Cooling and Heating
Natural Growth and Decay
According to Newtons 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

Torricellis Law
Water tank with hole
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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
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The Differential Equation
K a constant