Chapter 1 FirstOrder Differential Equations 1 Sec 1

Chapter 1: First-Order Differential Equations 1

Sec 1. 4: Separable Equations and Applications Definition 2. 1 A 1 st order De of the form is said to be separable. 1 2 3 3 2

Sec 1. 2 How to Solve ? 3

Sec 1. 4: Separable Equations and Applications 1 3 2 4 Solve the differential equation It may or may not possible to express y in terms of x (Implicit Solution) 4

Sec 1. 4: Separable Equations and Applications Solve the IVP 5

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

Sec 1. 2 How to Solve ? Remember division 3) Remember division 7

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

Sec 1. 4: Separable Equations and Applications 1 3 2 4 Solve the differential equation It may or may not possible to express y in terms of x (Implicit Solution) 9

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Modeling and Separable DE Natural Growth and Decay The Differential Equation Cooling and Heating According to Newton’s Law of cooling K a constant serves as a mathematical model for a remarkably wide range of natural phenomena. q. Population Growth q. Compound Interest q. Radioactive Decay q. Drug Elimination Torricelli’s Law Water tank with hole 11

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? The Differential Equation K a constant 12

The Differential Equation K a constant 13