Section 2.6 Equations of Bernoulli, Ricatti, and Clairaut.

Slides:



Advertisements
Similar presentations
SECOND-ORDER DIFFERENTIAL EQUATIONS
Advertisements

Section 9.2 Systems of Equations
A second order ordinary differential equation has the general form
Differential Equations Separable Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.
Objective - To graph linear equations using x-y charts. One Variable Equations Two Variable Equations 2x - 3 = x = 14 x = 7 One Solution.
Solving Systems of Equations: Elimination Method.
Vocabulary: Chapter Section Topic: Simultaneous Linear Equations
Differential Equations 6 Copyright © Cengage Learning. All rights reserved. 6.1 Day
Modeling with differential equations One of the most important application of calculus is differential equations, which often arise in describing some.
3-D Systems Yay!. Review of Linear Systems Solve by substitution or elimination 1. x +2y = 11 2x + 3y = x + 4y = -1 2x + 5y = 4.
Ordinary Differential Equations
Section 4-1: Introduction to Linear Systems. To understand and solve linear systems.
Chapter 8 Systems of Linear Equations in Two Variables Section 8.2.
Section 2.5 Implicit Differentiation
Warm-up It’s as easy as 1-2-3! 1)Start by separating variables. 2)Integrate both sides. 3) Solve for dy/dx. Solve = k(y – 80) This represents Newton’s.
Differential Equations Separable Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.
Bernoulli Differential Equations
(1) The order of ODE: the order of the highest derivative e.g., Chapter 14 First-order ordinary differential equation (2) The degree of ODE: After the.
Differential Equations
Math 3120 Differential Equations with Boundary Value Problems Chapter 2: First-Order Differential Equations Section 2-5: Solutions By Substitution.
Section 5.3 Solving Systems of Equations Using the Elimination Method There are two methods to solve systems of equations: The Substitution Method The.
Solving Linear Systems by Substitution
Topic: U4L2 Solving Nonlinear Systems of Equations EQ: How can I solve a system of equations if one or more of the equations does not represent a line?
1 Chapter 1 Introduction to Differential Equations 1.1 Introduction The mathematical formulation problems in engineering and science usually leads to equations.
Differential Equations Linear Equations with Variable Coefficients.
Differential Equations MTH 242 Lecture # 05 Dr. Manshoor Ahmed.
Graphing Linear Equations 4.2 Objective 1 – Graph a linear equation using a table or a list of values Objective 2 – Graph horizontal or vertical lines.
Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.
Section 1.1 Basic Definitions and Terminology. DIFFERENTIAL EQUATIONS Definition: A differential equation (DE) is an equation containing the derivatives.
Section 11.1 What is a differential equation?. Often we have situations in which the rate of change is related to the variable of a function An equation.
Graphing Linear Equations
Algebra Review. Systems of Equations Review: Substitution Linear Combination 2 Methods to Solve:
First-order Differential Equations Chapter 2. Overview II. Linear equations Chapter 1 : Introduction to Differential Equations I. Separable variables.
Lesson 9.6 Topic/ Objective: To solve non linear systems of equations. EQ: How do you find the point of intersection between two equations when one is.
Section 9.4 – Solving Differential Equations Symbolically Separation of Variables.
Rewrite a linear equation
Chapter 10 Conic Sections.
First-Order Differential Equations
Equations Quadratic in form factorable equations
Basic Definitions and Terminology
Families of Solutions, Geometric Interpretation
Chapter 6 Conic Sections
Chapter 12 Section 1.
Example: Solve the equation. Multiply both sides by 5. Simplify both sides. Add –3y to both sides. Simplify both sides. Add –30 to both sides. Simplify.
Systems of Nonlinear Equations
Solving Nonlinear Systems
Solving Systems using Substitution
First order non linear pde’s
Linear Differential Equations
A second order ordinary differential equation has the general form
PARTIAL DIFFERENTIAL EQUATIONS
Linear Inequalities and Absolute Value Inequalities
Section 11.2: Solving Linear Systems by Substitution
Solving Linear Systems Algebraically
Solve a system of linear equation in two variables
Use power series to solve the differential equation. {image}
Solve the differential equation. {image}
Solve Linear Equations by Elimination
Systems of Linear Equations
Solving Systems of Equation by Substitution
Section 7.1: Modeling with Differential Equations
5.1 Solving Systems of Equations by Graphing
Use power series to solve the differential equation. y ' = 7xy
Systems of Equations Solve by Graphing.
Equations Quadratic in form factorable equations
Chapter 8 Systems of Equations
Example 2B: Solving Linear Systems by Elimination
2 Chapter Chapter 2 Equations, Inequalities and Problem Solving.
Solving a System of Linear Equations
PARTIAL DIFFERENTIAL EQUATIONS
Presentation transcript:

Section 2.6 Equations of Bernoulli, Ricatti, and Clairaut

BERNOULLI’S EQUATION The differential equation where n is any real number, is called Bernoulli’s equation. For n = 0 and n = 1, the equation is linear and can be solved by methods of the last section.

SOLVING A BERNOULLI EQUATION 1.Rewrite the equation as 2.Use the substitution w = y 1 − n, n ≠ 0, n ≠ 1. NOTE: dw/dx = (1 − n) y −n dy/dx. 3.This substitution turns the equation in Step 1 into a linear equation which can be solved by the method of the last section.

RICATTI’S EQUATION The nonlinear differential equation is called Ricatti’s equation.

SOLVING A RICATTI EQUATION 1.Find a particular solution y 1. (This may be given.) 2.Use the substitution y = y 1 + u and to reduce the equation to a Bernoulli equation with n = 2. 3.Solve the Bernoulli equation.

CLAIRAUT’S EQUATION The nonlinear differential equation y = xy′ + f (y′) is called Clairaut’s equation. Its solution is the family of straight lines y = cx + f (c), where c is an arbitrary constant. (See Problem 29.)

PARAMETRIC SOLUTION TO CLAIRAUT’S EQUATION Clairaut’s equation may also have a solution in parametric form: x = −f ′ (t), y = f (t) − t f ′ (t). NOTE: This solution will be a singular solution since, if f ″(t) ≠ 0, it cannot be determined from the family of solutions y = cx + f (c).