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Questions tagged [integrating-factor]

For questions about integrating factors in general as well as their application to solving ODEs.

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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}+...
mrbiggles's user avatar
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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 \...
The Math Potato's user avatar
3 votes
1 answer
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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 ...
HERO's user avatar
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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 $\...
potato420's user avatar
2 votes
2 answers
373 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 ...
Harikrishnan M's user avatar
1 vote
1 answer
21 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 ...
Thomas Christopher Davies's user avatar
2 votes
1 answer
117 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{\...
Jacob Ivanov's user avatar
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3 answers
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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, ...
Simón Flavio Ibañez's user avatar
-3 votes
1 answer
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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 ...
Euler007's user avatar
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2 votes
1 answer
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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)...
EE18's user avatar
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3 answers
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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 ...
cstar112's user avatar
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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 ...
amaphi's user avatar
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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 ...
Nero's user avatar
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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 ...
Raghav Madan's user avatar
1 vote
3 answers
135 views

How to solve $(y^2 +2x^2y)dx +(2x^3 -xy)dy=0$

I tried solving it by comparing it with $Mdx+Ndy =0$ for $M=y^2 +2x^2y$ and $N=2x^3 -xy$, $$\frac{\partial M}{\partial y}=2y+2x^2,~~~~\frac{\partial N}{\partial x}=6x^2-y$$ $\displaystyle\frac{\...
Gowhar1998's user avatar
2 votes
0 answers
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Separable Equations vs. Integrating factors when solving ODEs

I came across a problem that seemed to yield different solutions based on which method I try and employ in solving the ODE. I am guessing that it is due to some error on my side, but was wondering if ...
Tempra Tura's user avatar
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2 answers
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Two integrating factors giving two different equations

In the above question, we get two integrating factors 1/y² and t. These give rise to two different equations. Please explain the underlying phenomenon of why this happens.
Smriti Saxena's user avatar
1 vote
1 answer
109 views

Weird solution to seemingly simple PDE

Consider the following problem: \begin{equation} \begin{aligned} \frac{\partial}{\partial t} u(x, t)-\frac{\partial^2}{\partial x^2} u(x, t) & =1, \quad x \in \mathbb{R}, t>0 \\ u(x, 0) & =...
Incubu121's user avatar
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2 answers
171 views

Solving an implicit first-order differential equation

We have a problem stating: Solve $ y(6y^2-x-1)dx + 2xdy = 0 $ Since we can't simply separate the variables. Our theory state we can use theses formulas to find a factor that's only dependent on a ...
Void Dark's user avatar
1 vote
1 answer
519 views

Solving Simultaneous Differential Equations

I'm asking about ways to solve simultaneous differential equations. This is a question from my textbook. It asks to find $x$ and $y$ in terms of $t$: $$(1)\ \frac{dx}{dt}=3x+y$$ and $$(2)\ \frac{dy}{...
mimi's user avatar
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Solving SDE $dx_t = (A - a x_t) dt + (b) dZ_t$

I am new to stochastic differential equations. I would like to solve something like this: $dx_t = (A - a x_t) dt + (b) dZ_t$ where: $A = \frac{ - \delta k }{\delta + a} $ The solution is: $x_t = e^{-...
NC520's user avatar
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2 votes
2 answers
123 views

Why does this formula for an integrating factor for first-order ODE work?

The book Shaum's Outlines: Differential Equations, 3rd edition (page 33) provides the following condition (among others) for determining that a first-order ODE is amenable to solution via an ...
Alex D's user avatar
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How can the inexact differential equation $y' = \frac{1-\frac{1}{2}x^2y^2}{1+(x-\frac{1}{2}x^2)y^2}$ be solved?

Since this equation is of the form $y' = \frac{M(x,y)}{N(x,y)}$, I tried computing the partials $M_y$, $N_x$, $\frac{M_y - N_x}{N(x,y)}$ and $\frac{N_x - M_y}{M(x,y)}$ - unfortunately, all of these ...
appletax13's user avatar
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1 answer
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Solve inexact ODE with integrating factor

We have ODE $$0=-\frac{x}{y}+\frac{y}{x}y'$$ where $(x,y)\in]0,\infty[\times]0,\infty[$. It's inexact, since$$\frac{dP(x,y)}{dy}=\frac{x}{y^2}\neq-\frac{y}{x^2}=\frac{dM(x,y)}{dx}$$ I can only find ...
Malik's user avatar
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1 answer
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Integrating factor for DE of the form DE $f(xy) ydx+g(xy) xdy=0$

I know that IF is of the form $$ \frac{1}{Mx-Ny} $$ But I couldn't find a formal proof anywhere. So far I could do the following: $f(xy)x=P$ and $g(xy)y=Q$ $(\mu P)_y = (\mu Q)_x$ $f\mu + y f_y\mu+ ...
Dominic Joseph's user avatar
2 votes
2 answers
63 views

Solving $2(y+e^x)dy + (y^2+4y e^x)dx = 0 $ and understanding integrating-factors

We want to solve $2(y+e^x)dy + (y^2+4y e^x)dx = 0 $ which, across the spectrum is the standard format of the integrating factor technique for ODE. My book, however, covers only the integrating factors ...
algevristis's user avatar
2 votes
3 answers
115 views

Solve the differential equation: $(3x^2-y^2)dy-2xdx=0$

I have the differential equation: $$(3x^2-y^2) dy - 2x dx=0 $$ I need it to look like the equation: $$y'+p(x)y=q(x)$$ in order to apply the integrating factor. I believe it can be solved using ...
user112167's user avatar
0 votes
2 answers
62 views

How to solve ordinary differential equations: $({2xy^2 - y}){dx} + ({y^2 + x+ y}){dy} = 0$?

I used the integrating factor for this equation, when i supposed that it only have ${x}$ or ${y}$, i found that the integrating factor have $x$ and $y$:$$$$ Call $u(x,y)$ is integrating factor, we ...
Nguyễn Đức Bá's user avatar
1 vote
1 answer
102 views

formula for integrating factor that depends only on x (ODE problem)

If$\dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}}{N}=\psi\left( x\right)$ then ODE $Mdx+Ndy=0$ has integrating factor that depends only on x s.t.$\mu Mdx+\mu Ndy=0$ is exact. ...
Pixel Book's user avatar
2 votes
0 answers
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Obtaining the solutions of the original problem from the solutions to the associated Sturm-Liouville problem

According to Sturm-Liouville's theory, any second order linear ordinary differential operator $L=P(x)\dfrac{d^2}{dx^2}+Q(x)\dfrac{d}{dx}+R(x)$ can be reduced to the Sturm–Liouville form, $\mathcal{L}=\...
Invenietis's user avatar
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1 answer
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Linear equation integrating factor method

I want to ask simple questions. $$ dy/dx + p(x)y=q(x)$$ In courses, I see always $$p(x)y$$ but what if $$ p(x)y^2$$ or $$p(x)y^3$$ or more ? It doesn't change anything I guess but I want to ask anyway....
Tryingtogetsome's user avatar
0 votes
1 answer
49 views

Discrepancy in the integrating factor of a first-order linear differential equation

I'm writing down the differential equation that I was solving, but it should be relatable to any other first-order linear differential equation. The equation is: $x \frac{dy}{dx}=x^2+3y$ where $x>0$...
Strong Lizard's user avatar
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1 answer
60 views

First order diferential equation with two integrating factors can be solved by separation of variables

Consider the diferential equation $$M(x,y)dx+N(x,y)dy=0$$ if $f(x)$ and $g(y)$ are integrating factors then the diferential equation can be solved by separation of variables. I don’t know how to prove ...
Carlos V. Ramírez Ibáñez's user avatar
1 vote
1 answer
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Need help for finding an integrating factor that makes a differential exact and solving it

When finding the integrating factor for $$e^x(x+1)dx+(ye^y-xe^x)dy=0$$ I used $$\frac{N_x-M_y}{M}=\frac{xe^x-e^x}{e^x(x+1)}$$ and got $\frac{x-1}{x+1}$. Then my integrating factor when solving for $dy$...
Pooe's user avatar
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1 vote
1 answer
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How to Find $u(x,y)$ for the PDE $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$ using Method of Characteristics and Method of Integrating factors?

$$dx=\dfrac{dy}{2}=\dfrac{du}{(-2x+y)u+2x^2+3xy-2y^2}$$ $$\dfrac{dx}{dy}=\dfrac{1}{2} \implies x=\dfrac{y}{2}+A$$ By sagemath software, $$\dfrac{du}{dy}=\dfrac{(-2x+y)u+2x^2+3xy-2y^2}{2}=1.0 \, A^{2} ...
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1 answer
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Integrating Factors derivation confusion

I'm confused about equation 11 in the first image (the second is added for context). How did they determine the left half of the equation and where did the right half come from? It looks like the ...
Jonathon Ashoul's user avatar
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1 answer
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Finding the general solution by integrating factors -- where am I going wrong (example problem included)?

I'm taking a college course in differential equations after a long break from math classes, and I'm struggling with finding general solutions through integrating factors. Here is the "easy" ...
Robin's user avatar
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2 votes
2 answers
407 views

Proving that a function $\mu (x,y)$ is an integrating factor of a first order homogeneous ODE.

The question is from Differential Equations by SL Ross Book: Show that if the equation $$M(x,y)dx +N(x,y)dy=0 \tag{A}\label{A}$$ is homogeneous and $M(x,y)x +N(x,y)y \neq 0$, then $\frac1{[M(x,y)x +N(...
Aman Kushwaha's user avatar
1 vote
2 answers
103 views

An application of integrating factors (basic ODE)

I'm trying to use integrating factors to solve the ODE ($g_1, g_0$ are both functions of $\xi$): \begin{equation} \frac{d g_{1}}{d \xi}-\left(\frac{g_{0}^{\prime \prime}}{g_{0}^{\prime}}\right) g_{1}=-...
Albert's user avatar
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2 votes
0 answers
106 views

Solving $(y+\sin x \cos^2(xy)-2x^2y)dx+(x+\sin y \cos^2(xy))dy=0$

They ask you to solve the following differential equation, taking as data its integrating factor $u(x,y)$: $$(y+\sin x \cos^2(xy)-2x^2y)dx+(x+\sin y \cos^2(xy))dy=0; \quad u(x,y)=u(xy)$$ This is my ...
ANDRES FRANCISCO YUPANQUI PELA's user avatar
1 vote
2 answers
31 views

Solving $v'(x) - ie^{x/2} v(x) = e^{2ie^{x/2}}$

While considering the first order inhomogenous ODE $$ v'(x) - ie^{x/2} v(x) = e^{2ie^{x/2}}, $$ I became curious about the most concise way to produce a solution. For the homogenous equation $$v_h'(x)-...
Talmsmen's user avatar
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2 votes
0 answers
57 views

What is the integrating factor of this differential equation?

I've been trying to solve this differential equation: $$\frac{y}{x^2}(2+x^3y)dx-\frac{1}{x^2}(1-2x^3y)dy=0$$ The methods I've tried to produce the integrating factor for it don't seem to be working. ...
Ayibatari Ibaba's user avatar
3 votes
2 answers
106 views

What is the integrating factor for this non-exact differential equation?

I am trying to solve this non-exact differential equation: $$2y(x^2-y+x)dx\,+\,(x^2-2y)dy = 0$$ Assuming that the integrating factor is of the form $x^my^m$: $$2(n+1)x^{m+2}y^n-2(n+2)x^my^{n+1}+2(n+1)...
Ayibatari Ibaba's user avatar
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0 answers
97 views

Integrating Factor and Absolute Values: Any shortcuts? For example $y'+y/t=t$

I'm a little puzzled about solving 1st order linear ODEs and determining when absolute values can be dropped. To give a specific example consider $y'+\frac{1}{t}y=t$. With the integrating factor ...
Fractal20's user avatar
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0 votes
1 answer
316 views

Integrating factor of the form $x^ay^b$ for a non-exact ODE

The task is to find $a$ and $b$ such that $u=x^ay^b$ is an integrating factor (IF) for the ODE $ydx+x(xy-1)dy=0$. Attempt: For $u$ to be an IF for the given ODE, the multiplication of ODE by the IF ...
User32563's user avatar
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2 votes
1 answer
93 views

What is the solution on the interval (-3, 3) for this ODE?

I took Differential Equations a few years ago and am reading back through my old book in a little more detail, especially with regard to intervals of validity for solutions. I found one of the example ...
Skater1's user avatar
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1 vote
0 answers
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Factoring an integral.

I have a result of an integral where I need to be able to find integer solutions once a C is specified: $$F(a,b,z,y) = abzy + (a+y) \cdot (b+z) + C$$ My idea is as follows: find a $C'$ that allows ...
Alex's user avatar
  • 2,469
2 votes
2 answers
66 views

Integrating Factor of $(x\ln(y) + xy)\mathrm{d}x + (y\ln(x) + xy)\mathrm{d}y$

Last day I try to prove that the equation $$ (x\ln(y) + xy)\mathrm{d}x + (y\ln(x) + xy)\mathrm{d}y =0 $$ is not exact, really easy task, but I want to go further, I tried to find a integrating factor ...
Lepton Cat's user avatar
0 votes
1 answer
319 views

How to convert a differential equation to an exact form with use of an integrating factor?

I need to solve the differential equation $1+(x/y-\sin y)y'=0$. The equation is not exact so I try to use an integrating factor to make it exact. $u(x) =$ integrating factor, only dependent on $x$ (...
Marco's user avatar
  • 71
1 vote
2 answers
2k views

Solve $(x^2-y^2)dx+2xy\ dy=0$

I'm at the beggining of a differential equations course, and I'm stuck solving this equation: $$(x^2-y^2)dx+2xy\ dy=0$$ I'm asked to solve it using 2 different methods. I proved I can find integrating ...
Alejandro Bergasa Alonso's user avatar