Linear Equations; Method of Integrating Factors

1. Overview

What is Integrating factor? To sum it all up its basically a function that is chosen to make easier to solve  a given equation that has differentials included. It’s usually used to answer differential equations.. At first you may think the procedure is a bit odd but until you get towards the end is when it can make more sense. Examine a usual differential equation that we need to solve to find out how the variable y depends on the variable x. if we notice that the equation happens to be in first order then the highest derivative involved is a first derivative. Note that if it’s also a linear equation then this means that each term can involve y either as the derivative dy/dx OR through a single factor of y .

 2. Example

 An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integral. We begin to spot when it can be used. We begin with starting from a standard form of ordinary differential equation.

(dy)/(dx)+f(x)y=q(x),

Once we get our equation we can jot down that f(x) and g(x) are two random functions of x only. When handling equations like this you must know that this form isn’t separable BUT we can join together the two terms on the left side into a single differential by using an integrating factor.

First we compute the integrating factor IF which is given to us by integrating f(x) then expanding the answer:

IF=e^f(x)

Now we defined F(x)=f(x)dx. When we integrate the function f(x) to get F9X) we don’t add a constant. We are only focus in the function in the integrating factor F(X).

(dF/dx)=f(x)

This translates to the differential of the integrating factor:

(d/dx)e^F(x)=(dF/dx)x(e^F(x))=f(x)e^F(x)

Where we used the chain rule. We begin to multiply both side of the equation:

(e^F(x))(dy/dx)+f(x)e^F(x)y=g(x)e^F(x)

Here is where you may thing its getting pretty odd, but there’s a reason for that is that the left hand side is now simply the differential of the integrating factor multiplied by y:

d/dx(e^F(x)y)=(e^F(x))(dy/dx)+f(x)e^F(x)y

This now gives us a chance to write out the original equation in a simpler format:

d/dx(e^F(x)y)=g(x)e^F(x)

Now we begin to integrate both sides

d/dx(e^F(x)y)dx=g(x)(e^F(x))dx

Integration is the opposite of differentiation, which makes it possible for the left side to cancel each other out, we then get the following:

e^F(x)y=g(x)(e^F(x))dx

if the integration on the right side is easy to compute this will lead towards our solution.

 3. Videos

 I included some videos that solve different equations using the method I provided above.

Video #2

In this particular video the instructor is solving the following equation of

(dy/dx)+3y=x

-By: Robert Morel

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My Partner Rahman Hasanuzzaman went along to review Linear Equations

Theory:

 In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

Consider an ordinary differential equation that we wish to solve to find out how the variable y depends on the variable x. If the equation is first order then the highest derivative involved is a first derivative. If it is also a linear equation then this means that each term can involve y either as the derivative  OR through a single factor of y. Any such linear first order can be re-arranged to give the following standard form:

 dy/dx + P(x)y = Q(x);

Where P(x) and Q(x) are functions of x, and in some cases may be constants.

A linear first order o.d.e. can be solved using the integrating factor method. After writing the equation in standard form, P(x) can be identified. One then multiplies the equation by the following “integrating factor”:

 IF= e^integral (P (x)dx )

 This factor is defined so that the equation becomes equivalent to:

 d/dx (IF y) = IF Q(x),

Whereby integrating both sides with respect to x, gives:

IF y = Integral (IF Q(x) dx)

 

Finally, division by the integrating factor (IF) gives y explicitly in terms of x, i.e. gives the solution to the equation.

Video 1:

Video 2: