Questions tagged [integrating-factor]
For questions about integrating factors in general as well as their application to solving ODEs.
153 questions
1
vote
1
answer
67
views
Derivation of the integrating factor
Is this a valid derivation of the integrating factor?
\begin{align}&\frac{d\mu }{dx}=\mu P(x) \longrightarrow \frac{1}{\mu }d\mu =P(x)dx\longrightarrow \int \frac{1}{\mu }d\mu =\int P(x)dx\\\...
0
votes
1
answer
57
views
Interval of Definition of a Linear ODE Solution
For the linear ODE $x\frac{dy}{dx}-4y=x^6e^x$ my diff eq textbook says $\mu{(x)}=e^{-4\ln(x)}$ instead of $\mu{(x)}=e^{-4\ln|x|}$ because $P(x)=-\frac{4}{x}$ and $f(x)=x^5e^x$ are defined for $(0,\...
4
votes
1
answer
190
views
How to solve this differential equation $(\frac{2}{y} - \frac{y}{x^3})dx + (\frac{1}{xy} - \frac{2}{x^3})dy = 0$?
I am trying to solve the following differential equation:
$(\frac{2}{y} - \frac{y}{x^3})dx + (\frac{1}{xy} - \frac{2}{x^3})dy = 0$
I checked for exactness and found that the equation is not exact. I ...
- 41
1
vote
1
answer
68
views
determining limits with integrating factor method with double definite integrals
I have an integrating factor such as,
$$
IF = \exp\left(\int_0^t f(u) + g(u) \, du \right),
$$
where it has been integrated with respect to t and had dummy variable u subbed in. The equation is then
$$...
- 35
1
vote
0
answers
68
views
Since linear ODEs have a general solution, when are integrating factors needed?
Linear homogenous ODEs with constant coefficients have straightforward general solutions; common classes of inhomogeneous have general solutions as well.
Integrating factors are only useful for linear ...
- 6,461
0
votes
0
answers
65
views
Conditions on Coefficients for a first Order ODE
Just want some expert advice, since I refuse to let ChatGPT become my math tutor. Consider the ODE:
$$
\frac{dy}{dx} = \frac{Ax+By+C}{Dx+Ey+F}
$$
Assume that $B\neq-D$, i.e., that the equation is not ...
- 632
0
votes
0
answers
53
views
Confusion about the solutions of non-exact differential equation using integrating factors
Suppose have a differential equation $M(x,y) + N(x,y)y'=0$ where $M_y \neq N_x$, we can use an integrating factor $\mu(x,y)$ to convert this equation to $\mu(x,y)M(x,y) + \mu(x,y)N(x,y)y'=0$ to make ...
- 21
0
votes
1
answer
60
views
Solving a differential equation using separation of variables vs Integrating Factor. Difference in answer.
Solving as a variable separable equation:
$y'+16xy=6x \rightarrow y'=x(6-16y) \rightarrow \frac{1}{6-16y}dy=x$
integrating we obtain (using u=6-16y, dy=du/-16):
$\frac{-1}{16}\ln(6-16y)=\frac{x^2}{2}+...
0
votes
1
answer
43
views
+/- when solving $y' + \frac{3}{x}\times y = 3x - 2$ via Integrating Factor Method
According to the internet, the solution to $y' + \frac{3}{x}y = 3x - 2$ is $y = \frac{3x^2}{5} - \frac{x}{2} + \frac{C}{x^3}$.
However, when I use the Integrating Factor Method, I get $\mu = e^{\int \...
- 318
3
votes
1
answer
97
views
Prove that two integrating factors define a solution.
I've been toiling away at this proof problem from Chapter 2.4 ending exercises of Differential Equations 3rd ed by Shepley L. Ross, but to no avail.
Show that if $\mu (x, y)$ and $v(x, y)$ are ...
- 511
-1
votes
1
answer
87
views
Prove that $\frac{1}{N^2 + M^2}$ is an integrating factor for $M(x, y) dx + N(x, y) dy = 0$ when $N_x$ = -$M_y$ and $N_y$ = $M_x$
Prove that $\frac{1}{N^2 + M^2}$ is an integrating factor for $M(x, y) dx + N(x, y) dy = 0$ when $N_x = -M_y$ and $N_y = M_x$
I tried to show that the equation $uMdx+uNdx=0$ is exact by showing that
$\...
- 33
2
votes
2
answers
422
views
An integral of a differential equation that's troubling me [closed]
I am facing a problem in differential equations.
$$(x-x^3)dy = (y+yx^2-3x^4)dx\tag{Question}$$
I am completely recognisant that I can use the linear differential equation form here, as I have shown ...
- 243
1
vote
1
answer
36
views
Proofing solution formula for first order ODEs with constant coefficients using integrating factor method
I want to show for an ODE of the form
$$y'=ay+b \tag{1}$$
with $a\neq0$, $b$ constants, has infinitely many solutions,
$$y(t) = ce^{at}- \frac{b}a \tag{2}$$
with $c \in \mathbb{R}$ using the ...
2
votes
1
answer
155
views
Integrating Factor for Vorticity Evolution
The Vorticity Evolution in 2D Cartesian Coordinates, assuming incompressibility, is as follows:
$$ \frac{\partial \omega}{\partial t} = \nu \left( \frac{\partial^2 \omega}{\partial x^2} + \frac{\...
- 339
0
votes
3
answers
174
views
Is there a solution for this non-linear ODE involving exponentials?
There is an non-linear ODE that pops out of the equations a lot when trying to solve the linear case of some second order ODE's.
The equation is this:
$$\ddot{y}+\dot{y}^2=y^2$$
It's easy to see that, ...
