# Questions tagged [integrating-factor]

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

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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.
1 vote
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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) & =...
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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 ...
1 vote
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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 $$(2)\ \frac{dy}{... • 13 0 votes 1 answer 17 views ### Solving SDE dx_t = (A - a x_t) dt + (b) dZ_t I am new to stochastic differential equations. I would like to solve something like this: dx_t = (A - a x_t) dt + (b) dZ_t where: A = \frac{ - \delta k }{\delta + a}  The solution is: x_t = e^{-... • 193 2 votes 2 answers 98 views ### Why does this formula for an integrating factor for first-order ODE work? The book Shaum's Outlines: Differential Equations, 3rd edition (page 33) provides the following condition (among others) for determining that a first-order ODE is amenable to solution via an ... • 357 0 votes 0 answers 81 views ### How can the inexact differential equation y' = \frac{1-\frac{1}{2}x^2y^2}{1+(x-\frac{1}{2}x^2)y^2} be solved? Since this equation is of the form y' = \frac{M(x,y)}{N(x,y)}, I tried computing the partials M_y, N_x, \frac{M_y - N_x}{N(x,y)} and \frac{N_x - M_y}{M(x,y)} - unfortunately, all of these ... 1 vote 1 answer 55 views ### Solve inexact ODE with integrating factor We have ODE$$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}$$I can only find ... • 153 0 votes 1 answer 44 views ### Integrating factor for DE of the form DE f(xy) ydx+g(xy) xdy=0 I know that IF is of the form$$ \frac{1}{Mx-Ny} $$But I couldn't find a formal proof anywhere. So far I could do the following: f(xy)x=P and g(xy)y=Q (\mu P)_y = (\mu Q)_x f\mu + y f_y\mu+ ... 2 votes 2 answers 51 views ### Solving 2(y+e^x)dy + (y^2+4y e^x)dx = 0  and understanding integrating-factors We want to solve 2(y+e^x)dy + (y^2+4y e^x)dx = 0  which, across the spectrum is the standard format of the integrating factor technique for ODE. My book, however, covers only the integrating factors ... • 937 0 votes 2 answers 57 views ### How to solve ordinary differential equations: ({2xy^2 - y}){dx} + ({y^2 + x+ y}){dy} = 0? I used the integrating factor for this equation, when i supposed that it only have {x} or {y}, i found that the integrating factor have x and y:$$$$Call u(x,y) is integrating factor, we ... 1 vote 1 answer 58 views ### formula for integrating factor that depends only on x (ODE problem) If\dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}}{N}=\psi\left( x\right) then ODE Mdx+Ndy=0 has integrating factor that depends only on x s.t.\mu Mdx+\mu Ndy=0 is exact. ... 2 votes 0 answers 45 views ### Obtaining the solutions of the original problem from the solutions to the associated Sturm-Liouville problem According to Sturm-Liouville's theory, any second order linear ordinary differential operator L=P(x)\dfrac{d^2}{dx^2}+Q(x)\dfrac{d}{dx}+R(x) can be reduced to the Sturm–Liouville form, \mathcal{L}=\... • 781 0 votes 1 answer 42 views ### Linear equation integrating factor method I want to ask simple questions.$$ 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$$or more ? It doesn't change anything I guess but I want to ask anyway.... 0 votes 1 answer 33 views ### Discrepancy in the integrating factor of a first-order linear differential equation I'm writing down the differential equation that I was solving, but it should be relatable to any other first-order linear differential equation. The equation is: x \frac{dy}{dx}=x^2+3y where x>0... 0 votes 0 answers 75 views ### Integrating factor in a homogeneous equation (From Boyce and Di Prima Book). Prove that if$$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)}$$Please help ... 0 votes 1 answer 49 views ### First order diferential equation with two integrating factors can be solved by separation of variables Consider the diferential equation$$M(x,y)dx+N(x,y)dy=0$$if f(x) and g(y) are integrating factors then the diferential equation can be solved by separation of variables. I don’t know how to prove ... 1 vote 1 answer 44 views ### Need help for finding an integrating factor that makes a differential exact and solving it When finding the integrating factor for$$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)}$$and got \frac{x-1}{x+1}. Then my integrating factor when solving for dy... • 11 1 vote 1 answer 56 views ### How to Find u(x,y) for the PDE u_x + 2u_y + (2x − y)u = 2x^2 + 3xy − 2y^2 using Method of Characteristics and Method of Integrating factors?$$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} ... 26 views

### 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 ...