Chapter 1: First-Order Differential Equations 1. Sec 1.4: Separable Equations and Applications Definition 2.1 1 A 1 st order De of the form is said to.

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Presentation transcript:

Chapter 1: First-Order Differential Equations 1

Sec 1.4: Separable Equations and Applications Definition A 1 st order De of the form is said to be separable

How to Solve ? Sec 1.2 3

12 4 Sec 1.4: Separable Equations and Applications 4 3 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 5 Solve the IVP

Implicit Solutions and Singular Solutions 6 Solve the IVP Implicit So, Particular, sol

How to Solve ? Sec Remember division 3) Remember division

Implicit Solutions and Singular Solutions 8 Solve the IVP Singular Sol division a general Sol Particular Sol Family of sol (c1,c2,..) No C a general Sol The general Sol Family of sol (c1,c2,..) 1)It is a general sol 2)Contains every particular sol Singular Sol no value of C gives this sol

12 4 Sec 1.4: Separable Equations and Applications 9 3 Solve the differential equation It may or may not possible to express y in terms of x (Implicit Solution)

10

11 Modeling and Separable DE 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 According to Newton’s Law of cooling Water tank with hole

12 The Differential Equation K a constant The population f 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?

13 The Differential Equation K a constant