Questions tagged [integrating-factor]
For questions about integrating factors in general as well as their application to solving ODEs.
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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 ...
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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}+...
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+/- 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 \...
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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 ...
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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
$\...
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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 ...
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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 ...
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Integrating Factor for Vorticity Evolution
The Vorticity Evolution in 2D Cartesian Coordinates, assuming incompressibility, is as follows:
$$ \frac{\partial \omega}{\partial t} = \nu \left( \frac{\partial^2 \omega}{\partial x^2} + \frac{\...
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Is there a solution for this non-linear ODE involving exponentials?
There is an non-linear ODE that pops out of the equations a lot when trying to solve the linear case of some second order ODE's.
The equation is this:
$$\ddot{y}+\dot{y}^2=y^2$$
It's easy to see that, ...
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Show that $\mu(x)$ is a integrating factor
Let $D \subseteq \mathbb{R}^{2}$ be a simply connected domain.
Furthermore, let $f, g: D \longrightarrow \mathbb{R}$ be two
continuously differentiable functions with $
\frac{\frac{\partial}{\partial ...
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Why is this the general solution of this DE?
I am reading a device physics text (Sze Physics of Semiconductor Devices, 3e, Chapter 2.4.3) and the author makes the claim that the solution $y(x)$ to a simple linear DE of the form
$$y' +P(x)y = Q(x)...
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Integrating factor of two variables
I am trying to solve the following:
$$
(x + x^2 + y^2) dy - ydx = 0.
$$
with an integrating factor involving both x and y. Indeed, it seems that an integrating factor of only one variable would not be ...
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A question about ODE
How to prove that $\frac{1}{xP(x,y)+yQ(x,y)}$ is an integrating factor of a homogeneous linear differential equation $P(x,y)$d$x+Q(x,y)$d$y=0$?
I have seen a proof:
Suppose that $Q \neq 0$, multiply ...
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Integrating factor having "two variables" in differential equations (first order DE)
Suppose we have the following problem:
$$(y-xy^2)dx+(x+x^2y^2)dy=0 \label{1}\tag{$*$}$$
If we try to find an integrating factor $\mu$ of a single variable (I.e., either $\mu(x)$ or $\mu(y)$) we will ...
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Integrating factor for non-exact ODE $y(1+2x^2+2y^2)\; \mathrm dx + x(1-2x^2-2y^2)\; \mathrm dy=0,$
If we have a non-exact ODE, then to convert it to an exact ODE we multiply the ODE with an integrating factor $\mu(x,y)$.
Lets us say we have the following ODE: $$M(x,y)dx+N(x,y)dy=0,$$ and let us ...
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