Linear Differential Equations | lesson 2

Показать описание

Linear equations - use of integrating factor

Consider the equation dy/dx + 5y = e²ˣ
This is clearly an equation of the first order , but different from those we have dealt with so far. In fact, none of our previous methods could be used to solve this one, so we have to find a further method of attack.

The above equation can be express in the form dy/dx + Py = Q , where P and Q are functions of x ( or constants ).
This equation is called a linear equation of the first order and to solve any such equation, we multiply both sides by an integrating factor which is always e^∫pdx
This converts the LHS into the derivative of the product of y and the integrating factor.

So to solve dy/dx + 5y = e²ˣ
where P = 5, therefore ∫pdx = 5x and the integrating factor is therefore e⁵ˣ
Note : In determining ∫pdx, we do not include a constant of integration. This omission is purely for convenience, for a constant of integration here would in practice give a constant factor on both sides of the equation, which would cubsequently cancel. This is one of the rare occasions when we do not write down the constant of integration.

So to solve dy/dx + 5y = e²ˣ we will multiply each term by the integrating factor
e⁵ˣ(dy/dx) + 5ye⁵ˣ=e⁷ˣ
The LHS can be express as d/dx(ye⁵ˣ)
Hence:
d/dx(ye⁵ˣ) = e⁷ˣ
By integrating both sides, we have
ye⁵ˣ= ⅐e⁷ˣ +c
and y = ⅐e²ˣ+ce⁻⁵ˣ

If you find this video interesting, kindly subscribe to my channel for more exciting Maths tutorials.

Рекомендации по теме

Комментарии

Professor Tambuwal, thank you for another great video/lecture on Linear Differential Equations.

georgesadler

Thank you very much you made maths easy for me
Big thanks

OkonkwoChristy

Very awesome video here. Have you got some videos on Riccati differential equations?

tomasito_

Can you do tutorials using D- operators

danielalelo

My God continue to bless you sir 🙌.... and please sir can you give us a tutorial on integration by parts please

ahmadsaifullahjalo

There is this calculation of D.E where y=vx, where v=y/x and dy/dx=v + xdv/dx. How is it different from dy/dx py = Q

amoszegrkaar

why can't we say sinxcosx is the same as cotx then proceed to integrate cotx

tlhalefoletshabo

Linear Differential Equations | lesson 2

First Order Linear Differential Equations

First Order Linear Differential Equation & Integrating Factor (introduction & example)

How to Solve First Order Linear Differential Equations

Introduction to Linear Differential Equations and Integrating Factors (Differential Equations 15)

Second Order Linear Differential Equations

Differential equation introduction | First order differential equations | Khan Academy

Differential equations, a tourist's guide | DE1

Finding particular linear solution to differential equation | Khan Academy

CUET PG Maths Ordinary Differential Equations Previous Year Questions | CUET PG Mathematics PYQs

Linear Differential Equations & Integrating Factors - Intro

Solving First-Order Linear Differential Equations - Introduction with Examples

Differential Equations: Lecture 2.3 Linear Equations

First order, Ordinary Differential Equations.

The Method of Integrating Factors for Linear 1st Order ODEs **full example**

Linear Differential Equations & the Method of Integrating Factors

Separable First Order Differential Equations - Basic Introduction

4 Types of ODE's: How to Identify and Solve Them

Calculus 2: Linear Differential Equations (Video #14) | Math with Professor V

Differential Equations: Lecture 1.1-1.2 Definitions and Terminology and Initial Value Problems

🔵15 - Linear Differential Equations: Initial Value Problems (Solving Linear First Order ODE's)...

Differential Equations Introduction | Differential Calculus Basics #differentialequation

How to solve ANY differential equation

Linear Differential Equations | lesson 2

How To Solve First Order Homogeneous Differential Equation

The Method of Integrating Factors for Linear 1st Order ODEs full example