Find integrating factor and solve the equation 2

In summary, the conversation discusses the process of integrating a differential equation to determine if it is exact. The first step is to find the integrating factor, which is calculated to be e^(-x^2/2). However, after using the integrating factor on the equation, it is still not an exact calculation due to a mistake in the differentiation. The correct result should be -1 instead of -x.
  • #1
naspek
181
0
Question:

(1 - x - z)dx + dz = 0

let M = (1 - x - z) ; N = 1

dM/dz = -x ; dN/dx = 0
hence, not exact.

integrating factor...

1/N[ dM/dz - dN/dx ] = 1[-x]
.......= -x

e^∫-x dx = e ^ [(-x^2)/2]


times integrating factor with DE..

{e ^ [(-x^2)/2] -x e ^ [(-x^2)/2] - z e ^ [(-x^2)/2]}dx +
e ^ [(-x^2)/2]dz = 0

dM/dz = -e ^ [(-x^2)/2] ; dN/dx = -xe ^ [(-x^2)/2]

the problem is, i calculated the integrating factor.. but, the equations still not the exact calculation.
 
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  • #2
naspek said:
let M = (1 - x - z)

dM/dz = -x

Did you forget how to differentiate?:wink:
 
  • #3
gabbagabbahey said:
Did you forget how to differentiate?:wink:

oh s#|t..!
it should be -1
thanks gabbagabbahey
 

1. What is an integrating factor?

An integrating factor is a function that is used to transform a linear differential equation into an exact differential equation, making it easier to solve.

2. How do you find the integrating factor for a differential equation?

To find the integrating factor, you need to first determine the form of the differential equation. Then, you can use various methods such as the method of variation of parameters or the method of undetermined coefficients to find the integrating factor.

3. Can you give an example of finding the integrating factor?

Sure! Let's say we have the differential equation dy/dx + 2y = 3x + 5. To find the integrating factor, we first identify the form of the equation as dy/dx + P(x)y = Q(x). In this case, P(x) = 2 and Q(x) = 3x + 5. Then, we use the formula e^(∫P(x)dx) to find the integrating factor, which is e^(∫2dx) = e^(2x).

4. How do you solve a differential equation using an integrating factor?

Once you have found the integrating factor, you can use it to transform the original differential equation into an exact differential equation. This will allow you to use methods such as separation of variables or the method of integrating factors to solve for the solution of the original equation.

5. Can you explain why finding the integrating factor is useful?

Integrating factors are useful because they allow us to solve certain types of differential equations that would be difficult or impossible to solve otherwise. By transforming the equation into an exact differential equation, we can use techniques that are already known for solving such equations, making the process easier and more efficient.

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