Solving (x^2+y)dx + (x+e^x)dy = 0 with Integrating Factor

In summary, the given differential equation cannot be solved using the methods mentioned in the conversation. The correct solution is not known and cannot be found using traditional methods.
  • #1

Homework Statement

From Elementary Differential Equation by Boyce and Diprima
Chapter 2 Miscellaneous Problems #11
(x^2+y)dx + (x+e^x)dy = 0


Homework Equations

multiplying an integrating factor to make the DE exact:
1. du/dx = u(My - Nx)/ N

2. du/dx = u(Nx-My)/ M

The Attempt at a Solution

First try: I guessed this can be changed into exact DE so, I tried with the two above equation:
equation 1 gave me:

du/u = e^x/(x+e^x)
I don't know how to solve this...

then equation 2 gave me:

u = e^((e^x)*ln(x^2+y))

I am not sure if multiply this integrating factor to the original DE will make it exact...

Second try: I manipulated the given DE and changed it to a linear form:

dy/dx = -(x^2+y)/(x+e^x)

dy/dx + 1/(x+e^x) * y = (-x^2)/(x+e^x)

and I found integrating factor to be:

I = e^∫1/(x+e^x) dx

which I am unable to solve...
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  • #2
Third try:

take integral of both side

∫(x^2+y)dx + (x+e^x)dy = ∫0

(x^3)/3 + yx + xy + ye^x = c //move constant c from left side to right side

(x^3)/3 + 2xy + ye^x = c

however it's not quite the same as the answer...

Wolframalfa does not solve this one!
  • #3
## (x^3/3)+xy+e^x=c ## cannot be the solution for ##(x^2+y)dx + (x+e^x)dy = 0##.

Differentiate: [tex]
d((x^3/3)+xy+e^x) = d(x^3/3) + d(xy) + d(e^x)
\\ = x^2dx + xdy + ydx + e^xdx
\\ = (x^2 + y + e^x)dx + xdy \ne (x^2+y)dx + (x+e^x)dy

1. What is the purpose of using an integrating factor in solving this equation?

The purpose of an integrating factor is to transform a non-exact differential equation into an exact one, which can then be solved using standard integration techniques.

2. How do you determine the integrating factor for this equation?

The integrating factor for this equation can be determined by multiplying both sides of the equation by the integrating factor, which is equal to the reciprocal of the coefficient of the term that is missing the independent variable. In this case, the integrating factor would be 1/(x^2+y).

3. Can this equation be solved without using an integrating factor?

Yes, this equation can be solved without using an integrating factor, but it requires more complex integration techniques such as the method of separation of variables or the method of substitution.

4. Are there any specific conditions that must be met in order to use an integrating factor for solving this equation?

Yes, the equation must be in the form of M(x,y)dx + N(x,y)dy = 0, where M and N are functions of x and y only. Additionally, the equation must not be exact, meaning that ∂M/∂y ≠ ∂N/∂x.

5. Can the integrating factor method be applied to other types of differential equations?

Yes, the integrating factor method can be applied to a variety of non-exact differential equations, including first-order linear equations and some second-order equations.

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