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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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Integrating Factor Counterexample

It has been my understanding that if there exist for an ODE in the form $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$ integrating factors that depend only on $x$, then the formula $u(x) = e^{-\int \frac{2D=curl\...
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Solving differential equation involving anti-symmetric part

I am looking for the steady state solution of a Fokker-Planck equation. The process involves a constant drift and position-dependent removal/insertion, thus leading to non-zero a steady state ...
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4answers
53 views

How to solve $\frac{dy}{dx}\ln(x)+\frac{2y}{x}=1$?

$$\frac{dy}{dx}\ln(x)+\frac{2y}{x}=1$$ The solution has evaded me thus far. I've tried pursuing the integrating factor technique (writing the equation as a differential and trying to find an ...
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1answer
43 views

Identifying best method for solving a $1^{st}$ order O.D.E

I am faced with this d.e: $$y' = \frac{2xy+3y^2}{2xy +x^2}$$ I know these methods for solving a $1^{st}$ order, o.d.e: *Integrating factor *Separation of variables *Exact d.e *Using intgrating ...
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2answers
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Solving $\frac{dy}{dx} = y - 3x$ with $y(-1) = 2$

Alright I know I'm asking for an answer with this, however on all of the examples presented to me with my notes in class and online, It doesn't show $y' = yx \leftarrow \textrm{example}$. It shows ...
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82 views

Integrating Factors Via Minimization

I am trying to algorithmize an idea to find integrating factors for inexact differential equations via optimization. Currently, I'm sticking to simple cases that can already be solved by other means. ...
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61 views

Does nudging an exact differential equation nudge or destroy the identity integrating factor?

This question will be related to this one, if for no other reason because a positive answer to the latter would likely help to solve the former. Consider the differential equation $(y)\ dx + (x)\ dy =...
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Integrating Factor for Closed but not Exact Differential Form

(I kind-of skipped ODEs in undergrad, so my knowledge is a little sketchy; please excuse me if this question is elementary.) (In addition, this problem is a bit of a sketch; I'll flesh it out with ...
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1answer
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How to find the integrating factor for this ODE?

$$(\cos^2x+y\sin2x)y'+y^2=0 \\ (\cos^2x+y\sin2x)dy+y^2dx=0 \\ \frac{\partial}{\partial x}(\cos^2x+y\sin2x)=-2\sin2x+2y\cos2x \\ \frac{\partial}{\partial y}y^2=2y $$ This equation is inexact, so I ...
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Solution of Ordinary Differential Equation(ODE) using Integrating Factor(I.F.)

Q) If $x(t)$ is a solution of $(1-t^{2})\;dx - tx\;dt = dt$ and $x(0)=1$, then $x\left ( \frac{1}{2} \right )$ is equal to (A) $\frac{2}{\sqrt{3}}\left ( \frac{\pi }{6}+1 \right )$ (B) $\frac{2}{\...
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$[\cos(x-y)+\sin(x-y)]dx + [\sin(x-y)-\cos(x-y)]dy$

the differential equation is given, integrating factor is as $\mu(x+y)$. find the integrating factor and solve the equation. $x-y\neq(2k+1)\pi /2$ $[\cos(x-y)+\sin(x-y)]dx + [\sin(x-y)-\cos(x-y)]dy$ ...
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1answer
55 views

Total differential equation with an integrating factor depending on the product $xy$

Description Show that if the quantity $$\frac{\frac{\partial P(x,y)}{\partial y}-\frac{\partial Q(x,y)}{\partial x}}{yQ(x,y)-xP(x,y)}$$ is a function $g(z)$ of the product $z=xy$, then the quantity: ...
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199 views

Non-Exact differential Equation

I am trying to solve a non-exact differential equation $\frac{s-b}{s}ds - \frac{b(2\beta-b)}{8{\beta}^2}db =0$ where $\beta >0$ is a constant. Boundary conditions are given by $b(s=0)=0$. A ...
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Integrating factor in Method of Characteristics

I have a PDE which takes the form: $$ \frac{\partial u}{\partial t} +c\frac{\partial u}{\partial x} = f(x,t) $$ So I understand how to get to the point where we make a coordinate change, $\xi = x-ct$...
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Exact Differential Equation Integrating Factor

Finding an integrating factor can be a genuine mathematical art. However, certain differential forms can remind us of differentiation techniques that may aid in the solution of the equation at ...
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1answer
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Solution of $y(3y+10x^2)dx-2x(y+3x^2)dy=0$

My attempt: I tried to find the Integrating Factor but couldn't find it by the standard methods. Also this is a non-linear differential equation which is non-homogeneous. I couldn't find any ...
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1answer
28 views

Implicit Integrating Factor

I have the ODE given below and am told to first find a suitable integrating factor to obtain an implicit solution $F(x,y)=C$ and then solving explicitly for $x$. $y - 3y^3 = \left(y^4 + 2x\right)y'$ ...
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2answers
124 views

Find a first integral of an ODE system

I have the system $$ \left\{\begin{array}{ccl} \dot x & = & 2xy \\ \dot y & = & x+y^2 \\ \end{array} \right. $$ and I need to find a first integral $H$ of the system. This is easy if ...
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A vector field with an integrating factor has a first integral

I have a problem that I don't know how to solve it, It says: Let $X=(f,g)\in \mathcal C^1(\mathbb{R})^2$ be a vector field and consider the system $\dot x = f(x,y), \; \dot y = g(x,y) $. If the ...
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1answer
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Help to solve this ODE by an integrating factor

I need help with the following ODE: $$ \frac{\partial S}{\partial t} - (a+b) S = -A(t)$$ The solution is supposed to be: $$S(t) = \int_{t}^{\infty}A(u) \> e^{-(a+b)(u-t)}du.$$ The integrating ...
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2answers
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differential Eq, how do I make it Exact? (using an integrating factor?)

$$ye^xdx-(4y+3e^x)dy=0$$ $$(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}) * \frac{1}{M}$$ $$e^{\int(4/y)}=y^4 $$ when multiplying $$y^4$$ though the problem still does not become ...
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4answers
159 views

Solving the non exact differential equation

Solve the following differential equation: $a(x\frac{dy}{dx}+2y)=xy\frac{dy}{dx}$. --Edited: see edit notes I am having trouble solving this equation, problems that I run into are outlined below. ...
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Exact Differential: Integrating Factor in Higher Dimensions

In two dimensions, we can turn every inexact differential $f(x,y)dx+g(x,y)dy$ into an exact version by multiplying both functions $f(x,y)$ and $g(x,y)$ by an additional function $I(z)$, i.e. $I(z) f(x,...
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How to solve this Linear Differential Equation?

This may look like a kind of homework question, but I am at a loss here. I came across this Linear Differential equation. I know how to solve LDE using Integrating Factor. And this equation is of the ...
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2answers
161 views

Deciding when to drop the absolute values in differential equation?

I am currently solving the following differential equation (link is to another post): $\dfrac{dr}{d \theta}+r\tan \theta =\frac{1}{\cos \theta}$ The following is in standard form (i.e. $\dfrac{dr}{d\...
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1answer
76 views

Differential equation integrating factor exact solution

I am trying to find an integrating factor for my differential equation \begin{equation} \left( c + \frac{1}{1-z}\right)g(z) + \frac{dg}{dz}\left(\frac{a}{1-z}z - 1\right)+\frac{a}{1-z} z\frac{df}{dz}=...
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2answers
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Basic math problem with Integrating Factors: Differential Equations

I was watching a youtube tutorial on Integrating Factors and I'm lost in a part where the derivative of: xy' + 1y becomes xy. Please I need some clarification on that part. Secondly. I was watching ...
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263 views

Find the integrating factor of $(x^2y-2xy^2)\,dx+(x^3-3x^2y)\,dy=0$

Find the integrating factor of the differential equation: $$(x^2y-2xy^2)\,dx+(x^3-3x^2y)\,dy=0$$ What I tried: This is a homogeneous equation. Therefore, $$I.F=\frac{1}{Mx+Ny}=\frac{1}{(x^2y-2xy^2)...
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1answer
117 views

Does Using an Integrating Factor give a General Solution? (Lin. ODE)

When solving a linear ODE of the form $$y' + p(x)y = q(x)$$ does using an integrating factor, $e^{\int p dx}$ provide us a method to get a general solution to the ODE, or is it a particular solution? ...
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Integrating factor mistake? [closed]

Can someone help me get the Integrating factor for this? $$y'\tan x+ y =e^{2x}\sin x$$ I got tansec²X But this was actually et multiple choice question and my teacher told me to retry the question as ...
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3answers
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Solving a ODE with integrating factor

$$(x^2 -1) \frac{dy}{dx} + 2xy = x$$ First I divided both sides by $(x^2 - 1)$ $$ \frac{dy}{dx} + {2x \over x^2 - 1}y = {x \over x^2 - 1} $$ and then I found the integrating factor $x^2 - 1$ and ...
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Solution of a Riccati ODE [duplicate]

I have the following ODE: $$\frac{d y}{d x}=y^2-(A \csc^2 x+B^2 \sin^2 x-C)$$ where $A,B,C$ are constants. Can one come up with an appropriate integrating factor to make it exact?
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Apostol's Calculus Vol 2, Chapter 10.20, Exercise 7

Let $\mu(x,y)$ be an integrating factor of the differential equation $P(x,y)\,dx+Q(x,y)\,dy=0$. I already showed that we have $$\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}=Q\frac{\...
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1answer
113 views

Solution for $\frac{dy}{dx}=\frac{1}{y^2}-\frac{1}{x^2}$, change of variables, integrating factor

I encountered the following nonlinear 1st order ODE as a simplified case of this question. $$\frac{dy}{dx}=\frac{1}{y^2}-\frac{1}{x^2}$$ I didn't find it in the literature and Wolfram Alpha couldn't ...
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2answers
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Integrating Factors Help

Studying for a test and I am a bit confused with this problem. $$y'+\frac y{(2xy-e^{-2y})}=0$$ It says that you can rewrite it as the following: $${(2xy-e^{-2x})}dy+ydx=0$$ I understand that you ...
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Finding an integrating factor $w$ for $\sigma$ such that $dw = \sigma$?

Suppose $\sigma = (yz + x^2z^2 + 3xy^2z)dx dy dz$ then how do we find a 2 - form $w$ such that $dw = \sigma$?. As I know that if it would have been $\sigma = \sum_{i} \alpha_{i} dx_{i}$ then as a ...
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4answers
87 views

What is $e^{\int{\frac{1}{x}dx}}$?

When doing integrating factors in order to solve differential equations, we often have something in a form similar to $$e^{\int{\frac{1}{x}dx}}$$ I know that $\int{\frac{1}{x}dx} = \ln{|x|}$, but for ...
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2answers
194 views

Make inexact DE exact (multiply by $\mu (x,y)$) and solve?

Show that the equation is not exact, but becomes exact when multiplied by the given integrating factor. Then solve the equation. $y'=e^{2x}+y−1$ with integrating factor $\mu(x,y)=1/xy^3$. I got ...
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0answers
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Show that this is a solution to the DE $y'+p(t)y=0$?

Here's the question: (b) Show that $y_1$ is a solution of the DE $y'+p(t)y=0$. (It is called a homogeneous DE, whenever $g(t) = 0$.) This is the second part to this problem: (a) Show that the ...
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1answer
50 views

Show/prove that the solution of a 1st order linear DE can be written in the following form?

Here's the problem: (1) (a) Show that the solution of the linear equation $y'+p(t)y=g(t)$ can be written in the form y=cy1(t)+Y(t), where c is an arbitrary constant. Identify the functions y1(t) ...
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Sufficient condition to have integrating factor a function of $x+y$

What will be the sufficient condition for the differential equation $M(x,y)\,dx + N(x,y)\,dy =0$ to have an integrating factor as a function of $(x+y).$ What will be integrating factor in that case? ...
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2answers
89 views

If $y' + p(x) y = q(x)$, $q \to 0$, and $p(x) \geq a > 0$, then $y \to 0$

Given the first order equation $$ y' + p(x)y=q(x) $$ where $p,q: \mathbb{R} \longrightarrow \mathbb{R} $ are continuous functions such that $p(x) \geq \alpha > 0, \forall x \in \mathbb{R}$ and $...
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1answer
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Concerning the solution to non homogeneous first order IVP

My teacher said that for first order non homogeneous IVP $$y'+a(x)y=f(x) , \ \ y(0)=y_o$$ we can get the solution directly using the following formula : $$\mu y|_{x=0}^x=\int_{\zeta=0}^{\zeta=x} f(\...
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1answer
307 views

Initial value problem with piecewise right hand side

I am not familiar at all with initial value problems that involve piecewise functions. The problem given is: Solve the initial value problem with piecewise right hand side $\left\lbrace\begin{array}{...
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3answers
77 views

Finding general solution to differential equation $y-3y^{7}=(y^{3}+6x)y'$

I have a question where I need to find the general solution of the differential equation $y-3y^{7}=(y^{3}+6x)y'$, where the solution is in the form $F(x,y)=C$. I am only concerned about finding $F(x,y)...
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3answers
42 views

Solving linear differential equation $y'+\frac{1}{3}sec(\frac{t}{3})y=4cos(\frac{t}{3})$ using integrating factor

Given \begin{array}{l} y^{\prime} +\dfrac{1}{3}\,\sec\left(\dfrac{t}{3}\right) y= 4\, \cos\left(\dfrac{t}{3}\right) \\ y(0)=3 \end{array} where $ 0<\dfrac{t}{3}<\dfrac{\pi}{2}$, I must find ...
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3answers
2k views

Finding integrating factor for non-exact differential equation $(4y-10x)dx+(4x-6x^2y^{-1})dy=0$.

I am given this equation $$(4y-10x)dx+(4x-6x^2y^{-1})dy=0$$ where I must find an integrating factor to turn this into an exact differential. The integrating factor is supposed to be in the form $\...
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3answers
67 views

Solving $\frac{dy}{dt} - y = 1 + \cos t$, using the integrating factor method

I'm having trouble trying to solve this differential equation: $\frac{dy}{dt} - y = 1 + \cos t$ So far, I've decided: $\mu(t) = e^{-1\int dt} = e^{-t}$ Which leads me to this: $e^{-t}\frac{dy}{dt} ...
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2answers
78 views

Integrating factor for a differential equation

Find the integrating factor for the equation: $(3x+\frac{6}{y})\mathrm{d}x+(\frac{x^2}{y}+\frac{3y}{x})\mathrm{d}y=0$. Write $P_1(x,y)=2x+\frac{6}{y}$, $Q_1(x,y)=\frac{x^2}{y}$, $P_2(x,y)=x$, and $...
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0answers
86 views

Integration of $e^\frac{-s^2}{4k}$

currently I'm trying to exercise function integrals and got stuck at a particular function which I found on the internet: $\int e^{\frac{-s^2}{4k}} ds$ Sadly, there is no solution online. This my ...