# Questions tagged [integrating-factor]

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

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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 ...
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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 ...
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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{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 ...
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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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### 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" ...
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 $\frac1{[M(x,y)x +N(... • 1,664 1 vote 2 answers 103 views ### An application of integrating factors (basic ODE) I'm trying to use integrating factors to solve the ODE ($g_1, g_0$are both functions of$\xi$): \frac{d g_{1}}{d \xi}-\left(\frac{g_{0}^{\prime \prime}}{g_{0}^{\prime}}\right) g_{1}=-... • 1,579 2 votes 0 answers 106 views ### Solving$(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 ... 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 $$v_h'(x)-... • 1,222 2 votes 0 answers 57 views ### What is the integrating factor of this differential equation? I've been trying to solve this differential equation:$$\frac{y}{x^2}(2+x^3y)dx-\frac{1}{x^2}(1-2x^3y)dy=0$$The methods I've tried to produce the integrating factor for it don't seem to be working. ... 3 votes 2 answers 106 views ### What is the integrating factor for this non-exact differential equation? I am trying to solve this non-exact differential equation:$$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)... 0 votes 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 ... • 1,509 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 ... • 832 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 ... • 67 1 vote 0 answers 100 views ### 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 ... • 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 ... 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$x$(... • 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 ...