Variation of parameters Definition and 116 Threads

Homework Statement Find a particular solution for these second order differential equations. Homework Equations 1) y''+9y=tan3t 2) y''+y=tan^2t The Attempt at a Solution I want to find a fundamental solutions y1 and y2 because I want to find a particular solution like this...

Homework Statement Solve for general solution with variation of parameter y'''(x) - y'(x) = x The Attempt at a Solution I initially looked at y'''(x) - y'(x) = x only and I foudn my answer to be y(x) = C_1e^{x} + C_2e^{-x} + 1 - x Now i looked through my book and it says it works for...

Given t^2 y'' -t(t+2)y' = (t+2)y= 2t^3 and y1= t, y2= te^t Find the particular solution- I ve worked the problem to [ -2t^2 -2t] by: -t * Integral [ 2t* te^t/ t^2e^t] + te^t * Integral [ 2t^2/ t^2e^t] whereas the book states that it is simply -2t^2. Can you guys tell me where I made...

Homework Statement \mathbb{x'}= \begin{bmatrix} 1 & 1 \\ 4 & 1 \end{bmatrix} \mathbf{x} \ + \ \begin{bmatrix} 2e^{t} \\ -e^{t} \end{bmatrix} Find the general solution. Homework Equations The Attempt at a Solution Well i found the eigenvalues of the matrix That i'll call...

ty''-(t+1)y'+y=t^2 I know I have to use variation of parameters to solve this. But I am stuck and cannot figure out how to get the homologous equation! y''-(1+\frac{1}{t})y'+\frac{1}{t}*y=t I don't know how to solve this homologous equation in this format. Is it R^2+(1+1/t)R+1/t = 0 ? How...

I am working on a problem requiring variation of parameters. When I calculated the wronskian, I got an answer, which differed from the book only by a "-" (mine was -, the book's was +). So I switched my functions for y1 and y2 and got the answer the book had. Is there a standard for which...

I have already read one thread on Lagrange's method of variation of parameters and it was very useful, but I am still confused about the use of the constraint. If the solution to the homogeneous second order equation contains two functions, with arbitrary constants: y= Ay1 + By2...

I am trying to solve a problem along the lines of y'' + 2y' + y = e^(-x) (2 + 1/x^2).. The actual one I am trying to solve differs slightly. I was trying to solve it using the method of variation of parameters.. However it is new to me and was too confusing. So first I get: y comlpiment...

Homework Statement Find a particular solution by method of variation of parameters: t2y'' - 2y = 3t2 - 1 given: y1 = t2 y2 = t-1 Homework Equations The Attempt at a Solution I get Y(t) = t^2ln(t) - \frac{1}{3}t^2 + \frac{1}{2} The book gives Y(t) = t^2ln(t) +...

I was looking through my DE book and a problem intrigued me. I eventually figured it out but I do not understand the logic. I was wondering if anyone here could help me out. The question says: Use the method of variation of parameters to show that...

Homework Statement 4y'' + y = cosx Solve using variation of parameters Homework Equations The Attempt at a Solution from a) -> yc(x) = c1cos(x/2) + c2sin(x/2) let y1 = cos(x/2) , y2 = sin(x/2) y1y2' - y2y1' = 1/2cosx/2 + 1/2sinx/2 = 1/2 u1' = ? How do I find this?

1.Homework Statement We know the derivation of the method of variation of parameters for second order scalar differential. The task is to derive the method of variation of parameters for scalar equations using this approach: first convert the scalar equation into the first order system and...

The equation of motion of a rocket with mass depletion during ascent and subject to drag forces can be written as M(t) dV/dt = A - M(t)g - BV^2 (Eq. 1) with initial condition V(t=0) = 0 (V is velocity and t is time) Let us assume a linear mass depletion according to...

Homework Statement Hello, i have a small problem regarding this questions, If the function vs(t) is a function for t>=0, i can solve thus no problem (we are required to solve using variation of parameters). now i have a small problem, its not about how to solve it ,but how to approach...

Homework Statement Find a particular solution using variation of parameters. y'' + 3y' + 2y = 4e^x Homework Equations yp = -y1 * INT (y2f(x)/W[y1,y2]) dx + y2 * INT (y1f(x)/W[y1,y2]) dx The Attempt at a Solution So, first I find the homogeneous solution, correct? r2 + 3r + 2 = 0, so...

Homework Statement What do you do if one of the roots to the characteristic equation of a differential equation is zero when using variation of parameters? Homework Equations The Attempt at a Solution The problem I encountered this in is y" - y' = 4t Characteristic equation r2 - r = 0 so...

Homework Statement Solve xy' - y = x3 (1) by using variation of parameters. The Attempt at a Solution Solving the homogeneous version of (1) gives yh = c1x Now we are to seek yp = A(x)*x (2) from (2) y'2 = A'*x +A plugging into (1) we have: x[A'*x + A] - Ax = x3...

Hi, I was solving the following second order ODE: http://www.texify.com/img/%5CLARGE%5C%21x%5E2%20y%5E%27%27-5xy%5E%27%2B5y%3Dx%5E6%20sinx.gif I used variation of parameters and found this solution...

Title should read "Combining", is there anyway a moderator could alter that so the search function isn't messed up? Homework Statement The Attempt at a Solution I am familiar with both methods, however combining the two is foreign to me. Anyone have any suggestions for this ODE? My...

Homework Statement You are given that two solutions to the homogeneous Euler-Cauchy equation x^2 \frac{d^2}{dx^2}y(x) - 5x \frac{d}{dx} y(x) + 5y(x) = 0 y1=x, y2=x^5 y''-\frac{5}{x}y'+\frac{5}{x^2}y=-\frac{49}{x^4} changing the equation to standard form use variation of parameters to find a...

Homework Statement Find the particular solution to the differential equation using method of variation of parameters: 4y''-4y'+y=16e^(t/2) The Attempt at a Solution Set 4y''-4y'+y=0 then the homogeneous solution is: y= c1*e^(t/2)+c2*te(t/2) set y1= e^(t/2), y2= te^(t/2)...

I have a question on the integration part of the Variation of Parameters. Given .y''+P(x)y'+Q(x)y=f(x) The associate homogeneous solution . y_c=c_1y_1 + c_2y_2. The particular solution . y_p=u_1y_1 + c_2y_2. u'_1 = -\frac{W_1}{W} = -\frac{y_2f(x)}{W} This is where I have question...

Homework Statement y''+25y=cot(5x) Find one possible solution The Attempt at a Solution I don't have any background in linear algebra, so I can't use cramers rule as a heads up, so I have to solve the system of equations (no linear algebra for this course is needed). Ok, so I take...

Homework Statement Using variation of parameters, find the general solutions of the differential equation Homework Equations y''' - 3''y + 3y' - y = et / t where et/t = g(t) The Attempt at a Solution I know how to solve these types of equations when its a second order, but I don't...

I am trying to solve the following equation using the variation of parameters method d2x/dt2-(q2Bz2/m2)x=qEx/m I have put x1=cos(t) and x2=sin(t) into the Wronskian method. Can someone tell me if these are the correct functions to use, or should I be using exponential functions. Any...

Homework Statement t²y"-t(t+2)y'+(t+2)y= 2t³ y1(t)=t y2(t)=te^t t>0 Homework Equations w(t)=y1*y2' - y1*y2 g=2t y=-y1∫(gy2)/w + y2∫(gy1)/w The Attempt at a Solution y1=t y1'=1 y2=te^t y2'=e^(t)+ te^(t) w(t)=te^(t)+t²e^(t)-te^(t)=t²e(t)...

Homework Statement Solve the problem: 4y'' - y = 8e^(.5t)/(2 + e^(.5t)) Homework Equations Particular solution of Y = X*integral(inverse of X multiplied by G) Finding eigenvalues and eigenvectors The Attempt at a Solution This might be a little too messy for anyone to make...

Homework Statement (x2+1)y"+(2-x2)-(2+x)y=x(x+1)2 given 2 associated homogeneous solution are: ex and 1/x Homework Equations this is a question from shaum's outline differential equations chapter on "variation of parameters"The Attempt at a Solution so here what i got... yh=C1ex+C2(1/x)...

Homework Statement Find a particular solution to: y''+4y=20sec(2t) Homework Equations The Attempt at a Solution y''+4y=0 r^2+4=0 r=+or- 2i So, yc(t) = Asin(2t) + Bcos(2t) yp(t)= -cos(2t) ∫ 10sin(2t)sec(2t)dt + sin(2t) ∫ 10cos(2t)sec(2t)dt = -10cos(2t) ∫...

Allright, I understand that we need two solutions to be able to apply the method like y_{1} and y_{2} Problem gives 1 of them or let's you find only that 1 solution. But I can't apply the method since I don't have the other solution. The method I know is: u_{1}'(x)y_{1}(x)+u_{2}'(x)y_{2}=0...

Homework Statement I have to solve the following differential equation by the "variation of parameters" method.Homework Equations \frac{dy}{dx}x +2y = 3x The Attempt at a Solution The associated homogeneous equation of the initial equation is: \frac{dy}{dx} = -2x^{-1}y So \frac{1}{y}dy =...

Homework Statement Given that x, x2 and 1/x are solutions of the homogeneous equation corresponding to: x^3y''' + x^2y''-2xy'+2y=2x^4 x>0 determine a particular solution. Homework Equations The Attempt at a Solution I'm trying to solve this problem using three...

Ok here's my problem: 1. Solve the inhomogeneous second order de: x^2y" - 3xy' + 4y =x^4 2. Worked: y(p) = 1/4*x^4 Given: y(1) = x^2 y(2) = log(x)*x^2 3. I just need help getting the roots of the given de so i can determine y(h) of this de. As...

Hey all, this is a little confusing, because the "variation of parameters" that I have been taught in class is different then what I find in most texts... I have y''' + y' = tan(x) Most textbooks use the wronskian and work from there, what I was taught to do is set it up as the...

Homework Statement http://img27.imageshack.us/img27/6083/variationofparametersfop.jpg

Homework Statement Find the general solution of the following diff. eqn. y''(t) + 4y'(t) + 4y(t) = t^(-2)*e^(-2t) where t>0 Homework Equations General soln - Φgeneral(t) + Φparticular(t) Wronskian - Φ1(t)Φ22'(t) - Φ2(t)Φ1'(t) The Attempt at a Solution I'm solving by...

Homework Statement y''+y=tan(x)+e^{3x}-1 Homework Equations homogeneous solution: y_{hom..}=C_{1}cos(x)+C_{2}sin(x) particular solution: y_{parti..}=v_{1}' cos(x)+v_{2}' sin(x) The Attempt at a Solution v_{1}' cos(x)+v_{2}' sin(x)=0 (1) -v_{1}' sin(x)+v_{2}' cos(x) =...

Hi, When using the method of variation of parameters to solve something like; y'' + y' = 2^x I got the aux. equation: r^2 - r =0 which gives the roots r=0,1 How do I find the complementary equation yc?

Consider, x' = x + 3y^3 y' = -3y I am trying to use the fundamental matrix, F(t), and 3y^3 as my g(t) in order to plug into the variation of parameters formula... Xp = F(t) * \integral{ F(t)^-1 * g(t) } , Am I going about this the wrong way? I am trying to get...

What is "Variation of Parameters"? Homework Statement None. General. Homework Equations I don't know. :( ? The Attempt at a Solution ? I am taking a class right now on engineering analysis (which I am finding it to be more like partial differential equations mixed with...

Homework Statement y''+2y'+y = 4t^2 - 3 + (e^-t)/t of course i evaluated the general soltuion to be c1e^-1t + c2te^-1t but now how do you do the right part? i tried y=At^2+Bt+c+1/(Dt+E)*e^-t as a solution but after differentiating it twice and putting it into the eqaution i got...

Question is attached as Clipboard01.jpg I have tried the use Variation of Parameters to solve this question, but I kept getting wrong answer. This is What I get y=(2e^x)(Cos(e^x))+0.5(e^(-x))Cos(e^(-x))-2Sin(e^(-x)) This is the right answer: y=-Sin(e^(-x))-(e^x)Cos(e^(-x)) Procedure is...

As the name suggest, this problem is an undetermined coefficients problems where variation of parameters is necessary to solve. As with my previous question; This is not a homework problem, but it is out of the textbook so I figured this would be the appropriate place to ask if I am doing it...

[SOLVED] Variation of Parameters Homework Statement y^(4)-6y^(3)=-5sinx The Attempt at a Solution I factored this at x^3(x-6)=0 so my r values are 0,6 also using for y(p) Dcosx + Esinx y=Ae^0 + Be^6x + Dcosx + Esinx ? y' =6Be^6x -Dsinx + Ecosx y'' =36Be^6x-Dcosx - Esinx...

Homework Statement Okay so when solving a system of D.E.s using Variation of Parameters I know that first I find the complementary solution Xc and then do a bunch a of crap after that using the fundamental matrix. Now I just came across a problem with repeated roots, so I just want to...

So I pretty much have this Differential Equation solved except that I have to integrate the expression \int \Phi(t)F(t)dt it has a star next to it in my attached work. Does this look readily integrable to anyone? For some reason nothing is ringing a bell. I suppose I could go by parts, but...

Solve by method of variation of parameters (x^2)y'' - (4x)y' + 6y = x^4*sinx (x > 0) Hey, I know how to solve problems using variation of parameters but only when the corresponding homogenous equation has constant coefficients... y'' - (4/x)y' + (6/x^2)y = 0.. the bit I am confused about...

Homework Statement Use the method of variation of parameters to determine the general solution of the given differential equation: y^(4) + 2y'' + y = sin(t) Homework Equations characteristic equation is factored down to (r^2 + 1)^2, so r = +/- i. this gives the general solution to be...

Within the description for the variation of parameters procedure is the restriction: y1u1' + y2u2' = 0. Can you explain this restriction, it is not obvious to me, I do not have an explanation where this comes from. Is it related to u[ \frac {dy}{dx} + P(x)y] = 0 from solving first...

Greetings, Regarding the two procedures: undetermined coefficients and variation of parameters, can both procedures be used interchangeably - meaning they both solve (non-homogeneous linear equations)? Does one method work better in certain situations, if so which method is preferred when...