Questions tagged [integrating-factor]

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

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How to solve this Differential equation(Integrating-factor)?

Can someone help me with this differential equation. $$y\tan(x) + \frac{e^{xy}}{\cos^2(x)} + x\tan(x)y' = 0$$ I guess it should be used integrating-factor but I can't solve it.
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48 views

Which is the correct integration using an integrating factor?

When shown equation $(1)$, I have two different answers for its integration, one mine, one more from a colleague and I am uncertain of which is the correct one. $$\left( \frac{\partial r}{\partial T}\...
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1answer
35 views

Solution of Inexact differential equation $x\sin(y)\,dy+(x^3-2(x^2)\cos(y)+\cos(y))\,dx$

$$x\sin(y)\,dy+(x^3-2x^2\cos(y)+\cos(y))\,dx$$ i tried solving the above d.e. the integrating factor comes out to be $e^{\int p\, dx}$ where $p$ was found out to be $x^2e^{-x^2}$. But the resulting d....
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24 views

Differential Equations Integrating y by x

This may be a bit of a silly questions, but when solving a differential equation by finding an integrating factor, is it possible to integrate a function of y and x by x? I understand that in multi ...
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1answer
37 views

Integrating the product of an exponential and a derivative

I have the following problem that I'm unsure how to tackle: $\frac{dm}{dt} = \frac{dn}{dt} - \lambda m$ I tried using the integrating factor method with IF = $e^{\lambda t}$ so I end up with: $me^{\...
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2answers
59 views

Finding an integration factor $x^ay^b$ to solve an ODE

I have to solve an ODE:$$2(y-3x)dx+x\left(3-\frac{4x}{y} \right) dy=0$$ I am given that I have to use the integrating factor x^a(y^b) where a and b are real numbers in order to turn the problem into ...
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How to find a non-exact ODE which becomes exact for a given integrating factor? [closed]

Do you have any non-exact differential equation example for the integrating factor $x + y$? I couldn't find any books.
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128 views

Second-Order In(exact) ODEs

The second total derivative of $F(x,\ y(x))$ is $F_y y'' + F_{yy}(y')^2 + 2F_{xy}y' + F_{xx}$. Thus by analogy to first-order exact ODEs, if one notices a second-order ODE where this pattern equals ...
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What is this integrating-factor approach to homogeneous ODE's good for?: Better for some equations? Illustrates something worthwhile?

We have an ODE of the form $$M(x,y)\,dx + N(x,y)\,dy = 0$$ in which both $M$ and $N$ are homogeneous functions of the same degree. The standard method for handling such an equation (assuming it's not ...
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26 views

Inexact differential and integrating factors

I am playing around with the differential $\mathrm{d}F = (x^2+4xy)\,\mathrm{d}x + (y^4+x^2)\,\mathrm{d}y$. This is not an exact differential, as $$\frac{\partial}{\partial y}(x^2+4xy)=4x$$ and $$\...
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Using an Integrating Factor to solve differential equation

I am trying to solve $$ (x^2y^3+y)dx+(x^3y^2-x)dy=0 $$ First of all we check this equation for exactness $\frac{\partial P}{\partial y}=3y^2x^2+1$ $\frac{\partial Q}{\partial x}=3x^2y^2-1$ The ...
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1answer
21 views

Why does this FO linear DE need an integrating factor?

This guy solves $$(t+y+1)\textrm{d}t - \textrm{d}y = 0$$ using an integrating factor $\mu(t)=e^{-t}$ and obtains $$y=-t-2+Ce^{-t}$$ The way I did it was more naive: $$\int \textrm{d}y = \int (t + ...
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32 views

Issue Using Integrating Factor

Let $F(x)$ be an arbitrary continuously differentiable function, and consider the following differential equation $$ y'+ \frac{F'(x)}{F(x)}y= \frac{1}{F(x)}.$$ If I use the Integrating Factor which ...
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1answer
22 views

Integrating Factor and $\frac{dy}{dx}+y=3x$

$\newcommand{\diff}{\frac{dy}{dx}}$ I was given the following problem: $$\diff+y = e^{3x}$$ My approach The integrating factor I found to be $P(x)=1$. Then my method to tackle the problem ...
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3answers
53 views

How should I approach this ivp problem with two differentiation about x?

The question given is as followed: $$x dy - y dx - (1-x²)dx = 0, \\ y(1)=1$$ How should I approach this question? I tried to start this problem by finding an integrating factor but what should I ...
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29 views

O.D.E Integrating Factor Help

I've been doing alright so far, but I can't seem to find the special integrating factor for this question, and Wolfram, as well as Symbolab, are unable to help. Please let me know: $$(2x^2y + x)dy + (...
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2answers
20 views

Why is this first-order ODE amenable to the method of integrating factor/separable?

Background We know that the first order differential equation in the form $$\frac{dy}{dx}+p(x)y=q(x)$$ is amenable to the method of integrating factor Question How is this first order equation ...
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29 views

Integration factor method with constants of integration

The question at hand is a step in solving the manifolds of a non-linear dynamical system. However the part I need help with is a step using the integrating factor method. We need to solve the 1st ...
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1answer
31 views

Transforming a partial derivative into a total derivative via integrating factor

Trying to solve an ordinary differential equation of the type: $y^{\prime}+P(x) y=Q(x)$ Wikipedia states: "To derive this, let $M_{(x)}$ be the integrating factor of a first order linear differential ...
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41 views

Common integrating factor of two differential equations.

The following differential equations have a common integrating factor. $$(3y+4xy^2)dx+(4x+5x^2y)dy=0$$ $$(6y+x^2y^2)dx+(8x+x^3y)dy=0$$ I have to find this integrating factor. I have arrived to ...
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5answers
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Determining integrating factor

Consider $$2x^2y\ln(y)+(x^3+x)y'=0$$ I want to determine the integrating factor $m(x,y)$ of the form $m(x,y)=m(xy)$. Let $a(x,y):=2x^2y\ln(y)$ and $b(x)=x^3+x$. So I want to find $m(x,y)$ such ...
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159 views

second order exact differential equation

I'm stuck in trying to solve the following ODE: $$y'' + 4xy' + (2+4x^2)y = 0$$ I tried finding an integration factor as if it was exact but I have not succeed. I also tried by trying to get rid of ...
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32 views

Proof dividing two integrating factors = k is solution to an exact equation

I have the next theorem: Let $P(x,y)dx + Q(x,y)dy = 0$ be a differential equation, and let $\mu _{1}(x,y)$ $\mu _{2}(x,y)$ two independent integrating factors for this differential equation. Then, $...
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2answers
71 views

Solve $y''+2y'-8y=e^{2x}$ using integrating factor method

I am trying to look for this question on the internet and Mathstack but I cannot find anything for second order inhomogeneous DE. And I need to know the solution using integrating factor method. $$y''...
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1answer
44 views

Find the integrating factor of IDE,

First of all, the question is not about the general solution. My question on my book asks me to find the integrating factor. Suppose I have simple IDE $$(\sin y)\mathbb dx +(\cos x)\mathbb dy=0$$ ...
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31 views

Solving ODE using integrating factor - ctmc

I am having trouble solving this first order ode using the integrating factor. Could anyone explain the solution in more detail?
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2answers
90 views

Finding an integrating factor for $(3t + \frac{6}{y})dt + (\frac{t^2}{y} + \frac{3y}{t})dy = 0$

I want to solve the differential equation $$ \begin{equation} (3t + \frac{6}{y})dt + (\frac{t^2}{y} + \frac{3y}{t})dy = 0\ \end{equation} $$ using an integrating factor of the form $t^a y^b$ in ...
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4answers
62 views

Integrating factor mistakes when solving 1 order ODE

I have an ODE: $$\frac{dy}{dx} + 3x^{2}y = x^{2}$$ . I got the following integrating factor: $$e^{x^3}$$ Then I multiplied both sides, but didn't come up with the right answer. It should be: $$...
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Integrating equations using the integrating factor with higher powers of y

I have often come across the following technique to integrate first order differential equations using the integrating factor: $$ \frac{dy}{dx}+P(x)y=Q(x) $$ $$ e^{{\int}P(x)dx}\frac{dy}{dx}+e^{{\int}...
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1answer
51 views

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
63 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
52 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
78 views

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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0answers
87 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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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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92 views

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
40 views

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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93 views

$[\cos(x-y)+\sin(x-y)]dx + [\sin(x-y)-\cos(x-y)]dy$

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
98 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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2answers
233 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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0answers
44 views

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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2answers
119 views

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
43 views

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
64 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
701 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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1answer
67 views

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
30 views

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
40 views

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 ...