Integrating Factor: Overview, Questions, Preparation

An integrating factor is a type of factor of a function in which a standard equation of differentiation can be multiplied for making the given function integral. It can be used with multivariable calculus. Also, when an inaccurate differential multiplies with an integrating factor, then it becomes an accurate differential. 

Explanation

The integrating factor method can be explained as the function selected in a particular order for solving the given differential equation. Mostly it is used in first-order ordinary linear differential equations. 

For instance, if the given differential equation is:
dz/dy + P(y) z = Q(y)

Then, the integrating factor will be defined as:

µ = e^{∫P(y) dy}

P(y) is the function of y, and multiple z and µ denote the equation's integrating factor. 

Steps for solving First Order Differential Equation by using an Integrating Factor

• Make a comparison of the given and differential equation to determine the value of P(y).
• Now, evaluate the µ, i.e., integrating factor.
• Multiply integrating factor and differential equation on both sides for getting equation like:
• µ dz/dy + µP(y)z = µQ(y)
• Now, at last, integrate the expression and get the desired result for the given equation: 
• µz = ∫µQ (y) dy +C.

Steps for Solving Second Order Differential Equation Using an Integrating Factor

• Let the second-order differential equation be: zm + P (y) z’ = Q (y)
• Now, substitute z' with t for making the equation similar to the first-order equation.
• After this, the equation can be solved with the steps explained above for the first-order differential equation. And, after performing the above steps, we will get an equation like μ t=∫μQ(y)dy+C
• Now, figure out the value of t by solving the equation. And integrate the equation for getting the required solution. 

Integrating factor is one of the important concepts that help Grade 12 students to solve differential equations. Here, they will learn all the concepts and methods of using integrating factors in calculus. 

1. Find the general solution of dz/dy = 3z/y+1 = (y + 1)⁴

Solution.

Firstly, find the integrating factor

I = e^{∫P dy} = e^{∫ -3/y+1} dy    

Now, ∫-3/ y+1 dy = -3 In(y+1)  = ln (y+1)^-3

Hence,  I = e^Ln(y+1)-3 = (y+1)^-3 = 1/(y+1)³

Now, multiply equation with I, then we get

1/(y + 1)³dz/dy – 3z/(y + 1)⁴ = (y + 1)

After integrating both the sides, we will get 

z/(y + 1)³ = ½ y² + y + C  where c is a constant.

Hence, the general solution is z = (y + 1)³ (1/2y² + y + C).

2. Solve integrating factor for the differential equation y^t +2ty=0

Solution:

Linear equation is of the form: dy/dx+ P(x)y= Q(x)

Now we, find the integrating factor

I.F.=e^∫Pdx

P=2t
I.F.=e^∫2tdt
=e^t2
Therefore, the integrating factor is e^t2 

3. Find the general solution of dq/dp +2pq = p
Solution.  q = ½ + ce^1/x2

Q. When can integrating factors be used?

Q: Is the integrating factor unique?

Q: How many ordered equations can be solved with integrating factor?

Q: Can integrating factors transform an inexact differential to an accurate one?

Q: Are integrating factors important for solving a differential equation?