Questions tagged [integrating-factor]
For questions about integrating factors in general as well as their application to solving ODEs.
131
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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.
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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) & =...
- 15
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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 ...
- 33
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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 $$(2)\ \frac{dy}{...
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Solving SDE $dx_t = (A - a x_t) dt + (b) dZ_t$
I am new to stochastic differential equations. I would like to solve something like this:
$dx_t = (A - a x_t) dt + (b) dZ_t$
where:
$A = \frac{ - \delta k }{\delta + a} $
The solution is:
$x_t = e^{-...
- 193
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2
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Why does this formula for an integrating factor for first-order ODE work?
The book Shaum's Outlines: Differential Equations, 3rd edition (page 33) provides the following condition (among others) for determining that a first-order ODE is amenable to solution via an ...
- 357
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How can the inexact differential equation $y' = \frac{1-\frac{1}{2}x^2y^2}{1+(x-\frac{1}{2}x^2)y^2}$ be solved?
Since this equation is of the form $y' = \frac{M(x,y)}{N(x,y)}$, I tried computing the partials $M_y$, $N_x$, $\frac{M_y - N_x}{N(x,y)}$ and $\frac{N_x - M_y}{M(x,y)}$ - unfortunately, all of these ...
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Solve inexact ODE with integrating factor
We have ODE $$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}$$
I can only find ...
- 153
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Integrating factor for DE of the form DE $f(xy) ydx+g(xy) xdy=0$
I know that IF is of the form $$ \frac{1}{Mx-Ny} $$
But I couldn't find a formal proof anywhere. So far I could do the following:
$f(xy)x=P$ and $g(xy)y=Q$
$(\mu P)_y = (\mu Q)_x$
$f\mu + y f_y\mu+ ...
2
votes
2
answers
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Solving $2(y+e^x)dy + (y^2+4y e^x)dx = 0 $ and understanding integrating-factors
We want to solve $2(y+e^x)dy + (y^2+4y e^x)dx = 0 $ which, across the spectrum is the standard format of the integrating factor technique for ODE. My book, however, covers only the integrating factors ...
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How to solve ordinary differential equations: $({2xy^2 - y}){dx} + ({y^2 + x+ y}){dy} = 0$?
I used the integrating factor for this equation, when i supposed that it only have ${x}$ or ${y}$, i found that the integrating factor have $x$ and $y$:$$$$
Call $u(x,y)$ is integrating factor, we ...
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formula for integrating factor that depends only on x (ODE problem)
If$\dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}}{N}=\psi\left( x\right)$ then ODE $Mdx+Ndy=0$ has integrating factor that depends only on x s.t.$\mu Mdx+\mu Ndy=0$ is exact.
...
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Obtaining the solutions of the original problem from the solutions to the associated Sturm-Liouville problem
According to Sturm-Liouville's theory, any second order linear ordinary differential operator $L=P(x)\dfrac{d^2}{dx^2}+Q(x)\dfrac{d}{dx}+R(x)$ can be reduced to the Sturm–Liouville form,
$\mathcal{L}=\...
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Linear equation integrating factor method
I want to ask simple questions.
$$ 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$$ or more ?
It doesn't change anything I guess but I want to ask anyway....
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Discrepancy in the integrating factor of a first-order linear differential equation
I'm writing down the differential equation that I was solving, but it should be relatable to any other first-order linear differential equation. The equation is: $x \frac{dy}{dx}=x^2+3y$ where $x>0$...
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Integrating factor in a homogeneous equation
(From Boyce and Di Prima Book). Prove that if $$M(x,y)\,dx+N(x,y)\,dy=0$$ is an homogeneous equation, then one of its integrating factors is:
$$ \mu(x,y)=\frac{1}{x\,M(x,y)+y\,N(x,y)}$$
Please help ...
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First order diferential equation with two integrating factors can be solved by separation of variables
Consider the diferential equation
$$M(x,y)dx+N(x,y)dy=0$$
if $f(x)$ and $g(y)$ are integrating factors then the diferential equation can be solved by separation of variables.
I don’t know how to prove ...
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Need help for finding an integrating factor that makes a differential exact and solving it
When finding the integrating factor for
$$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)}$$
and got $\frac{x-1}{x+1}$. Then my integrating factor when solving for $dy$...
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How to Find $u(x,y)$ for the PDE $u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2$ using Method of Characteristics and Method of Integrating factors?
$$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} ...
user817548
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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 ...
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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" ...
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answers
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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 $\frac1{[M(x,y)x +N(...
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An application of integrating factors (basic ODE)
I'm trying to use integrating factors to solve the ODE ($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}=-...
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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 ...
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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 $$v_h'(x)-...
- 1,114
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What is the integrating factor of this differential equation?
I've been trying to solve this differential equation:
$$\frac{y}{x^2}(2+x^3y)dx-\frac{1}{x^2}(1-2x^3y)dy=0$$
The methods I've tried to produce the integrating factor for it don't seem to be working. ...
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What is the integrating factor for this non-exact differential equation?
I am trying to solve this non-exact differential equation: $$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)...
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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 ...
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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 ...
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1
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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 ...
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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 ...
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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 ...
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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 $x$ (...
- 71
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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 ...
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why the integral of $g'(x)/g(x)$ is $\ln(|x|)$
To the integrating factor method for differential equations I have to make a step where the integral of $\frac{g'(x)}{g(x)}$ is $\ln(|x|)$.
I don't understand how this is derived.
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Solve differential equation by integrating factor
I have the differential equation $$2\frac{dy}{dx}+3y=e^{-2x}-5$$
I have determined that this needs solving using the integrating factor method.
My workings out are in the image provided.
Are my ...
- 23
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Trotter factorization
While reading a paper on the simulation of an exciton in potassium chloride using effective mass path integrals, a partition function is given as so,
$Z = ∫ dr_1⟨r_1∣e^{−βH}∣r_1⟩$
in which a symmetric ...
- 143
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2
answers
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Solution of first order inexact differential equation
I have following differential equation before me:
$(2y sin(x)+3y^4 sin(x)cos(x))dx-(4y^3 cos^2(x)+cos(x))dy=0$
It is an inexact differential equation. Its Integrating factor comes out to be $cos(x)$.
...
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Integrating factor for $xdx +(x-y^2)dy=0.$
How to find integrating factor for $xdx+(x-y^2)dy=0.$
Here $M=x$ and $N=x-y^2$. This equation is not exact. Clearly not seperable, homogeneous or linear(either in terms of $\frac{dy}{dx}$ or in terms ...
- 141
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1
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Is my solution to this first order ODE correct? (Integrating factor method)
The question wants me to prove the following ODE via integrating factor method:
$$ \frac{dx}{dt} + \frac{t}{(1+t^2)}x = \frac{1}{t(1+t^2)} $$
I got my integrating factor to be $ \mu(t)= \sqrt{1+t^2}$
...
- 97
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4
answers
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Solve $y’ = \frac{2-xy^3}{3x^2y^2}$ using integration factor [duplicate]
Show that the equation:
$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$
Has an integration factor that depends on $x$ And solve it that way.
Already we got to:
$$
y’ + \frac{xy^3}{3x^2y^2} = \frac{2}{3x^2y^2}
$$
...
- 1,647
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1
answer
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Solving a differential equation using Integrating Factor method
I need to to model a raindrop's velocity as it is falling with respect to time.
The assumptions made are that air resistance is negligible and that the raindrop is spherical
I was able to to calculate ...
- 43
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0
answers
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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.
- 11
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1
answer
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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.
$$\left( \frac{\partial r}{\partial T}\...
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1
answer
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Solution of Inexact differential equation $x\sin(y)\,dy+(x^3-2(x^2)\cos(y)+\cos(y))\,dx$
$$x\sin(y)\,dy+(x^3-2x^2\cos(y)+\cos(y))\,dx$$
i tried solving the above d.e. the integrating factor comes out to be $e^{\int p\, dx}$
where $p$ was found out to be $x^2e^{-x^2}$.
But the resulting d....
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1
answer
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Differential Equations Integrating y by x
This may be a bit of a silly questions, but when solving a differential equation by finding an integrating factor, is it possible to integrate a function of y and x by x? I understand that in multi ...
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Integrating the product of an exponential and a derivative
I have the following problem that I'm unsure how to tackle:
$\frac{dm}{dt} = \frac{dn}{dt} - \lambda m$
I tried using the integrating factor method with IF = $e^{\lambda t}$ so I end up with:
$me^{\...
- 5
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votes
2
answers
283
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Finding an integration factor $x^ay^b$ to solve an ODE
I have to solve an ODE:$$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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1
answer
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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.
- 11
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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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