What is Integrating factor: Definition and 123 Discussions
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.
Homework Statement
Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation:
5xy dy/dx = x2 + y2
Homework Equations
y=ux
dy/dx = u+xdu/dx
C as a constant of integration
The Attempt at a Solution
I saw a similar D.E. solved using the y=ux...
Homework Statement
Find an integrating factor of the form ##x^Ay^B## and solve the equation.
##(2y^2-6xy)dx+(3xy-4x^2)dy=0##
Homework Equations
##M=2y^2-6xy##
##N=3xy-4x^2##
##IF = exp(\int \frac{M_y-N_x}{N}\,dx)##
or
##IF = exp(\int \frac{N_x-M_y}{M}\,dy)##
The Attempt at a Solution
[/B]...
Homework Statement
##cos t \frac{dv}{dt} + (sin t) t = \frac{GM}{b^2 }\sin^3 t ##
Homework Equations
above
The Attempt at a Solution
im pretty stuck to be honest. It almost looks like a product rule on the LHS but it has the wrong sign, RHS I've tried writing ##sin^3 t## as ##(1-cos^2t)\sin...
2000
$\tiny{206.q3.2}\\$
$\textsf{3. use the method of integrating factor}\\$
$\textsf{to find the general solution to the first order linear differential equation}\\$
\begin{align}
\displaystyle
\frac{dy}{dx}+5y=10x
\end{align}
$\textit{clueless !}$
Homework Statement
Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentration of salt 2 grams of salt per lter is added at a rate of 3 liters per minute. The tank is mixed well and is drained at 3 liters a minute. How long does the process have to continue until...
I tried to put it in standard form as (dx/dy)+4x(1/(1+y))=y/(y+1). I get that the integrating factor is (y+1) but i am not sure if i am doing it right or what am I suppose to do next? I get (y+1) because the integrating of 1/(y+1) is ln(y+1) and since it has e, then ln cancels and i am left...
Homework Statement
hello, I was reading through the textbook and I have a hard time to understand this part:
Homework EquationsThe Attempt at a Solution
haven't been dealing with derivatives for a while, i don't understand how it got ln |u(t)| from the first equation.
Am I treating the...
While solving non-homogenous linear ODEs we make use of the integrating factor to allow us to arrive at a solution of the unknown function. Same applies to non linear ODEs where the ODEs are converted to exact differentials.
But what I don't understand is how and why would someone have come up...
Hello PF! We were doing mass balances on mixing tanks in one of my ChemE courses, and in one of the problems we arrived at the following DE:
\frac{dC_B}{d \theta} + 0.025C_B=0.0125 e^{-0.025 \theta}
Where CB is the concetration of salt in the tank and θ is time. The professor made us solve the...
Homework Statement
##C_{B}## is a function of ##\tau'##, and ##k_{1}##,##k_{2}##, and ##C_{A0}## are constants. I want to solve this differential equation
\frac {dC_{B}}{d \tau'} + k_{2}C_{B} = k_{1}C_{A0}e^{-k_{1} \tau'}
Homework EquationsThe Attempt at a Solution
Using the integrating...
Homework Statement
Find the general solution to the indicated equation:
cos(x)y' + ysin(x) = 1
Homework Equations
e^\int p(x)\,dx * y(x) = int\ f(x)\, dx e^\intp(x)\,dx + C
The Attempt at a Solution
Ok, I am having trouble getting started with this problem because I am not...
Homework Statement
Use an integrating factor to determine the general solutions of the following differential
equation:
dx/dt - 2/t = 2t3 + (4t2)(e4t)
Homework Equations
R(x) = e∫P(x).dx
The Attempt at a Solution
Usually the equation is in the form dx/dt + P(x)t = Q(x) but...
$
kxy \frac{dy}{dx} = y^2 - x^2 \quad , \quad
y(1) = 0
$
My professor suggests substituting P in for y^2, such that:
$
P = y^2
dP = 2y dy
$
I am proceeding with an integrating factor method, but unable to use it to separate the variables, may be coming up with the wrong integrating factor ( x )
Ok, so I have this differential equation.
\[(3x^2y+2xy+y^3)+(x^2+y^2)y\prime=0\]
First I needed to check to see if it is exact.
\(M=3x^2y+2xy+y^3\)
\(N=x^2+y^2\)
\(\dfrac{\partial M}{dy}(3x^2y+2xy+y^3)=3x^2+2x+3y^2\)
\(\dfrac{\partial N}{dx}(x^2+y^2)=2x+0\)
For the integrating factor, I...
Problem:
xy'+2y=3x
Attempt:
Divide by x...
y'+\frac{2y}{x}=3
I think I find the integrating factor by doing:
e^{\int \frac{2}{x}dx}
Not sure if that's right but if it is then the solution to the integral is just 2x.
Any help is appreciated
Homework Statement
Solve the initial value problem:
$$sin(x)y' + ycos(x) = xsin(x), y(2)= \pi/2$$
Homework Equations
The Attempt at a Solution
Recognizing it as a Linear First-Order Equation:$$\frac{dy}{dx}+y\frac{cosx}{sinx}=x$$
$$P(x)=\frac{cosx}{sinx}$$
Integrating...
Homework Statement
y' + (2/t)y = (cost)/(t^2), and the following condition is given: y(pi) = 0Homework Equations
The Attempt at a Solution
After employing the integrating factor, I find the solution to be:
y=e^{-2t} \int e^{2t} \frac{\cos(t)}{t^2} dt.
Evidently, this simplifies all the way to...
Here is the question:
Here is a link to the question:
Differential equations math question? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Homework Statement
Consider the general linear homogeneous second order equation:
P(x)y'' + Q(x)y' + R(x)y = 0 (1)
We seek an integrating factor μ(x) such that, upon multiplying Eq. (1) by μ(x), we can write the resulting equation in the form
[μ(x)P(x)y']' + μ(x)R(x)y = 0...
Homework Statement
Use parametrisation first, derive the equation including y and p = \frac{dy}{dx} and use the integrating factor method to reduce it to an exact equation. Leave the solution in implicit parametric form.
(y')^{3} + y^{2} = xyy'The Attempt at a Solution
I'm really lost at...
Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great.Consider the equation \frac{dy}{dx} + 5y = e2x
In this case, we begin multiplying both sides by e5x. This gives...
Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great.
Consider the equation \frac{dy}{dx} + 5y = e2x
In this case, we begin multiplying both sides by e5x. This...
Let the Wronskian between the functions f and g to be 3e^{4t}, if f(t) = e^{2t}, then what is g(t)?
So the Wronskian setup is pretty easy
W(t) = fg' - f'g = 3e^{4t}
f = e^{2t}
f' = 2e^{2t}
So plugging it in I would get:
e^{2t}g' - 2e^{2t}g = 3e^{4t}
Which results in
g' - 2g...
Homework Statement
Find an integrating factor for the first order linear differential equation
\frac{dy}{dx} - \frac{y}{x} = xe^{2x}
and hence find its general solution
Homework Equations
The Attempt at a Solution
I found the integrating factor which is e^{-lnx} = x^{-1}
and...
Homework Statement
I'm having a hard time to solve the following DE using an integrating factor, I'm asked to find one and solve the DE.
(x+y^2)-2xyy'=0.Homework Equations
The Attempt at a Solution
If I call u=u(x) (I assume the integrating factor depends only on x, not on y), the following...
Homework Statement
From Elementary Differential Equation by Boyce and Diprima
Chapter 2 Miscellaneous Problems #11
(x^2+y)dx + (x+e^x)dy = 0
ANS:(x^3/3)+xy+e^x=c
Homework Equations
multiplying an integrating factor to make the DE exact:
1. du/dx = u(My - Nx)/ N
2. du/dx = u(Nx-My)/ M
The...
xdy-ydx=(x2+y2)2(xdx+ydy)
(Hint: consider d(x2+y2)2)Homework Equations
d(x2+y2)2
d(arctan(y/x))=xy|-y/(x2+y2)The Attempt at a Solution
The answer is arctan(y/x)-(.25(x2+y2)2)=C
I correctly solved other problems of this type, but this is the hardest one in the problem set and I have no idea how...
Homework Statement
I have this statement:
If M(x,y)dx+N(x,y)dy=0 is a homogeneous DE, then μ(x,y)=\frac{1}{xM+yN} is its integrating factor. The problem is, how do we derive this integrating factor?
Homework Equations
For homogeneous DE, we have f(kx,ky)=k^n*f(x,y)
We also have...
Homework Statement
Solve the differential equation: t^2 y' + y^2 = 0
The Attempt at a Solution
Now, it's definitely possible to solve this via separable of variables. But I am curious to know if I can solve it with an integrating factor. Having done some reading, I noticed that this...
Hello,
I am solving an equation using integrating factor. I have come up to a specific point which is $$\dfrac{d}{dt} P_{02}(t) \cdot e^{(\lambda_3+\mu_3)t}=\lambda_2 \cdot P_{01}(t) \cdot e^{(\lambda_3+\mu_3)t}$$
from the previous equation, I have found $$P_{01}(t)=\lambda_1 \int_0^t...
Homework Statement
Hi, I attempted to solve this question, but it did not work well. I would be grateful if you could help me.
Show that an equation of the form xrys(mydx + nxdy)= 0 has an infinite number of integrating factors of the form xayb, and find expressions for a and b...
Hey,
I've just been following this proof for a integrating factor of (xy),
http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html
it starts at at equation (22)
I understood it all a few days ago and now I seem to have forgotten this one step.
It says...
Homework Statement
e∫^P(x)
∫\frac{x-2}{x(x-1)}dx
The Attempt at a Solution
so i split it into
∫\frac{x-2}{x(x-1)}dx
= ∫\frac{2x-1}{x^2-x}dx - ∫\frac{x+1}{x^2-x}dx
= ln(x2-x) - ∫\frac{x}{x^2-x} - ∫(x2-x)-1
= ln(x2-x) - ln(x-1) - ∫(x2-x)-1
ok. having problems working out...
I am really struggling with proving a ODE by means of using the integrating factor method.
My original problem was a Laplace transform
q'+2q=5sin(t) where q(0)=0
I believe i have got the correct naswer for this as being:- q= e^-2t +2sint-cost
I just need to confirm this i have my...
Homework Statement
(3(x^2)y + 2xy + y^3)dx + (x^2 + y^2)dy = 0
The Attempt at a Solution
This is from my notes, so I already have the answer. I just don't understand the very last step with the integrations.
(3(x^2)y + 2xy + y^3)dx + (x^2 + y^2)dy = 0
My = 3x^2 + 2x + 3y^2
Nx = 2x...
Hi all,
I am doing some Laplace Transforms as part of my HND, i have got an answer for this question
q' +2q = 5sint q(0)=0, t(0)=0
But i need to prove it by means of using an integrating factor method.
My original answer is:-
e^-2t +2sint-cost does this look right?
I also have...
Homework Statement
Find the general solution of the given differential equation cosxy'+(sinx)y=1
The Attempt at a Solution
I divided everything by cosx and got : y'+(tanx)y=secx
then after doing e to the integral of tanx i got : ∫d/dx[secx*y]=∫secx
after integrating and...
alright guys, I've been trying to tackle this for a couple of hours now.
dy/dt-2y=4-t
my integrating factor is e^(-2t) of course.
dy(e^(-2t))/dt-2ye^(-2t)=4e^(-2t)-te^(-2t)
then I get completely lost. how do I integrate when it's like this? My book simplifies the above equation into...
I keep struggling to find a solution to this IVP. We are supposed to use integrating factors
y'-(1/t)y=8t^2+te^t
t>0, y(1)=6
I get an integrating factor of (1/t) and general solution of y=4t^3+te^t+c but then i get e+2 for c. This doesn't seem correct to me, any suggestions?
This is the ODE: y' + siny + xcosy + x = 0.
The problem is: Find an integrating factor for the ODE above.
You can see my solution to the ODE here: https://www.physicsforums.com/showthread.php?t=543662. from my solution it seems that e^x(sec^2(y/2)) must be an integrating factor. but I fail...