1. Chapter 1: First-Order Differential Equations 1

2. Sec 1.4: Separable Equations and Applications Definition 2.1 1 A 1 st order De of the form is said to be separable. 2 3 3 2

3. How to Solve ? Sec 1.2 3

4. 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)

5. Sec 1.4: Separable Equations and Applications 5 Solve the IVP

6. Implicit Solutions and Singular Solutions 6 Solve the IVP Implicit So, Particular, sol 2 2 -2

7. How to Solve ? Sec 1.2 7 Remember division 3) Remember division

8. 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

9. 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)

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11. 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. 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. 13 The Differential Equation K a constant