Differential Equations

Slides:

Advertisements

Similar presentations

Equation of a Tangent Line

Advertisements

DIFFERENTIATION & INTEGRATION CHAPTER 4.  Differentiation is the process of finding the derivative of a function.  Derivative of INTRODUCTION TO DIFFERENTIATION.

Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.

Copyright © 2008 Pearson Education, Inc. Chapter 10 Differential Equations Copyright © 2008 Pearson Education, Inc.

1 6.3 Separation of Variables and the Logistic Equation Objective: Solve differential equations that can be solved by separation of variables.

7/4/2015 Cauchy – Euler’s Equations Chapter /4/2015 Cauchy – Euler’s Equations Chapter 5 2.

7/6/2015 Orthogonal Functions Chapter /6/2015 Orthogonal Functions Chapter 7 2.

Chap 1 First-Order Differential Equations

Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.

AP Calculus Ms. Battaglia

1Chapter 2. 2 Example 3Chapter 2 4 EXAMPLE 5Chapter 2.

Chapter 2 Solution of Differential Equations

Method Homogeneous Equations Reducible to separable.

1 Part 1: Ordinary Differential Equations Ch1: First-Order Differential Equations Ch2: Second-Order Differential Equations Ch3: The Laplace Transform Ch4:

Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin.

Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

3.5 – Solving Systems of Equations in Three Variables.

Application of Derivative - 1 Meeting 7. Tangent Line We say that a line is tangent to a curve when the line touches or intersects the curve at exactly.

Implicit Differentiation

Orthogonal Trajectories

Chapter 7 Determine the Relationship of a System of Equations 1/4/2009 Algebra 2 (DM)

Differential Equations Copyright © Cengage Learning. All rights reserved.

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.

3.1 Solving equations by Graphing System of equations Consistent vs. Inconsistent Independent vs. Dependent.

Differential Equations Chapter 1. A differential equation in x and y is an equation that involves x, y, and derivatives of y. A mathematical model often.

Math 3120 Differential Equations with Boundary Value Problems Chapter 2: First-Order Differential Equations Section 2-5: Solutions By Substitution.

AP Calculus Ms. Battaglia. Equations are separable if all x terms can be collected with dx and all y terms with dy. The solution procedure is called separation.

Differential Equations Linear Equations with Variable Coefficients.

Warm Up. Solving Differential Equations General and Particular solutions.

Chapter 21 Exact Differential Equation Chapter 2 Exact Differential Equation.

Advance Engineering Mathematics – Active Learning Assignment Topic: First ODE(Integrating Factor to Orthogonal trajectories of curves) First ODE(Integrating.

Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.

DO NOW: Find the inverse of: How do you find the equation of a tangent line? How do you find points where the tangent line is horizontal?

2( ) 8x + 14y = 4 -12x – 14y = x = x = 4 8x + 14y = 4 8(4) + 14y = y = y = -28 ___ ___ y = -2 The solution is (4, -2)

Unit 2 Lesson #3 Tangent Line Problems

Section 9.4 – Solving Differential Equations Symbolically Separation of Variables.

Lesson 3-6 Implicit Differentiation. Objectives Use implicit differentiation to solve for dy/dx in given equations Use inverse trig rules to find the.

Differential Equations

§ 4.2 The Exponential Function e x.

6.1 – 6.3 Differential Equations

Differential Equations

Differential Equations

Specialist Mathematics

Section 10.1 Separable Equations I

Chapter 12 Section 1.

Implicit Differentiation

MTH1170 Differential Equations

Solve the differential equation. {image}

Solve the equation for x. {image}

Calculus II (MAT 146) Dr. Day Monday, March 19, 2018

Solving Linear Systems Algebraically

Systems of Linear First-Order Differential Equations

Differential Equations

Solve the differential equation. {image}

Implicit Differentiation

Tangents and normals Remember, the tangent to a curve at a point is a straight line that just touches the curve at that point. The normal to a curve at.

Implicit Differentiation

9.1/9.3 – Differential Equations

Differential Equations

Problem: we can’t solve the differential equation!!!

Numerical Solutions of Ordinary Differential Equations

Separation of Variables and the Logistic Equation (6.3)

Boundary Value Problems

Section 8.2 Part 2 Solving Systems by Substitution

Section 6.3 Day 1 Separation of Variable

Solving a System of Linear Equations

Intersection Method of Solution

Reading Between the Lines!

Use Graphs of Functions

Presentation transcript:

Differential Equations Chapter 3 Application of First order Linear Differential Equations

Orthogonal trajectories Orthogonal Trajectories Chapter 3 Chapter 3

Orthogonal Trajectories Chapter 3

Orthogonal Trajectories Chapter 3 METHOD Eq.1. Given family of curve STEP 1 Find the differential equation for the given family of curves, by differentiating Eq.1. Eq.2. Orthogonal Trajectories Chapter 3 STEP 2 Find the differential equation of the Orthogonal Trajectory Eq.3. STEP 3 Solution of Eq.3. will be the equation of the family of orthogonal trajectories

Orthogonal Trajectories Chapter 3 Orthogonal Curves (1) Example Solution Orthogonal Trajectories Chapter 3 Two curve intersect perpendicularly if the product of the slopes of the tangents at the intersection point is -1. The differential equation for the orthogonal family of curves.

Orthogonal Trajectories Chapter 3 Orthogonal Curves (2) Example Solution (cont’d) Orthogonal Trajectories Chapter 3 The figure on the right shows these two orthogonal families of curves.

Orthogonal Trajectories Chapter 3 [1/2] Example: Find the equation of the orthogonal trajectories of the family of curves Eq.1. Solution: STEP 1 Differentiating Eq.1 Orthogonal Trajectories Chapter 3

Orthogonal Trajectories Chapter 3 [2/2] STEP 2 Differential equation of the orthogonal trajectories is Solving the differential equation by method of separation of variables STEP 3 Orthogonal Trajectories Chapter 3 is required equation of orthogonal trajectories .

Orthogonal Trajectories Chapter 3 Example: Find the equation of the orthogonal trajectories of the family of curves Eq.1. Solution: STEP 1 Differentiating Eq.1 we obtained Orthogonal Trajectories Chapter 3 STEP 2 The differential equation of the Orthogonal Trajectory is STEP 3 Solving this Homogeneous DE, we obtained the equation of family of orthogonal trajectories

Orthogonal Trajectories Chapter 3 Example: Find the equation of the orthogonal trajectories of the family of curves Eq.1. Solution: Differentiating Eq.1 we obtained STEP 1 Orthogonal Trajectories Chapter 3 STEP 2 The differential equation of the Orthogonal Trajectory is STEP 3 Solving this Homogeneous DE, we obtained the equation of family of orthogonal trajectories

Orthogonal Trajectories Chapter 3