Integrating factor Definition and 113 Threads

$\begin{array}{rl} \textit{Find } \mu(x): &\mu(x) =\exp\left(\int \dfrac{1}{x}\,dx\right)=e^{\ln{x}}=x\\ \textit{multiply thru by x} &xy^\prime+y=3x\cos 2x\\ \textit{rewrite as } &(xy)'=3x\cos 2x \\ \textit{}integrate &xy=\int 3x\cos 2x \...

I'm not sure if that is the right way to solve this excersice. Can someone maybe help and tell me if this calculation proofs the statement ?

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...

So I've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one) appreciate the help

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...

x2 + y + y2dx - x dy = 0 Integrating factor, I(x,y) = -1 / (x2 + y2) How to find the integrating factor ?Why I cannot use below method to solve the ode ? (1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR (1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)

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 Solve ##{dy/dx}-2xy=2x##Homework EquationsThe Attempt at a Solution Let ##P= -2x ## and Q= 2x, Integrating factor =## e^{-x^2} ## ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## ##y.e^{-x^2}={x^2} e^{-x^2}+∫ 2{x^3} e^{-x^2}dx## since ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## then...

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...

Homework Statement is my integral R(x) and R(y) correct ? why it look so complicated ? Homework EquationsThe Attempt at a Solution

How do you solve x ln(x) dy/dx = xe^x using the integrating factor? So far, I have put it into standard form. dy/dx + y/(xln(x))=xe^x/(x(ln(x))

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 https://webwork.utpa.edu/webwork2_files/tmp/equations/2d/02a7e6a06f5b2424758fa01cc965f71.png with https://webwork.utpa.edu/webwork2_files/tmp/equations/80/81c176aa8964438a63eb096513245f1.png Homework Equations [/B] Standard form: y' + p(x)y = f(x)...

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...

$$w'+2w=0\\ \frac { dw }{ dx } =-2w\\ I(x)={ e }^{ 2x }\\ \frac { dw }{ dx } { e }^{ 2x }=-2w{ e }^{ 2x }\\ \int { \frac { dw }{ dx } { e }^{ 2x } } dx=\int { -2w{ e }^{ 2x } } dx$$ Not sure what to do next.

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 (3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0

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 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...

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...

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...

Homework Statement Find an integrating factor for: xdy - (y + x^2 + 9y^2)dx = 0 Homework Equations P(x,y)dx + Q(x,y)dy = 0 Δμ = μyP - μxQ. where μ is the integrating factor. The Attempt at a Solution well, I don't know what I should do. I can use the formula I wrote but that would...

My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...

I am having trouble with the below: [ 4* (x^3/y^2) + (3/y)] dx + [3*(x/y^2) +4y]dy=0 I found My= -8x^3y^-3 - 3y^-2 and Nx= 3y^-2 i then subtracted Nx from My and divided by [3*(x/y^2) +4y] [-8x^3y^-3 - 6y^-2] / [3*(x/y^2) +4y]. can you guys give me a hint as to where my error is?

Hi, I have a general question regarding the integrating factor of first-oder linear DEs. All textbooks that I've seen (which aren't too many) simply drop the absolute symbol when the factor has the form exp(ln(abs(x))). This would evaluate to abs(x), yet the books use simply x. Why is that...