What is Variation of parameters: Definition and 116 Discussions

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations.
Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa.

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  1. P

    Inhomogeneous equations: Variation of Parameters

    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...
  2. F

    Understanding Variation of Parameters for Solving Differential Equations

    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...
  3. N

    Why is my particular solution not matching with the book's answer?

    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...
  4. B

    Variation of parameters ODE what am i doing wrong?

    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...
  5. V

    How Do You Solve the Homogeneous Equation for ty''-(t+1)y'+y=0?

    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...
  6. C

    Question on variation of parameters - ODE

    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...
  7. R

    Variation of parameters and the constraint

    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...
  8. R

    How Can I Use Variation of Parameters to Solve Differential Equations?

    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...
  9. P

    Variation of Parameters problem

    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) +...
  10. X

    Solution to a DE using variation of parameters

    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...
  11. B

    Variation of Parameters Nonhomogeneous Differential Equation

    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?
  12. R

    Deriving Variation of Parameters for Systems

    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...
  13. R

    Extension of Variation of Parameters to First Order Non-Linear ODE?

    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...
  14. S

    Approaching a Step Function Problem with Variation of Parameters

    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...
  15. A

    Differential equations - variation of parameters

    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...
  16. C

    Quick Question on Variation of Parameters Differential Equations

    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...
  17. Saladsamurai

    Variation of Parameters on a 1st Order DE

    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...
  18. J

    Mathematica and Variation of Parameters

    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...
  19. G

    ODE: Combining Undetermined Coeff. & VOP Method

    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...
  20. C

    ODE using variation of parameters

    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...
  21. B

    Differential Equations: Variation of Parameters

    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)...
  22. Y

    Question on Variation of Parameters

    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...
  23. L

    Variation of Parameters, system of equations

    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...
  24. M

    Differential equations, variation of parameters

    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...
  25. H

    Solving EM Problem with Variation of Parameters

    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...
  26. E

    Variation of Parameters (Diffy Equ.)

    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)...
  27. A

    Variation of parameters (Kinda having trouble with the integral)

    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...
  28. R

    Solving Linear Systems Using Variation of Parameters

    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)...
  29. J

    Finding a particular solution for y''+4y=20sec(2t)

    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) ∫...
  30. A

    Method of Variation of Parameters

    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...
  31. B

    Solving a first order linear differential equation by variation of parameters

    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 =...
  32. Wellesley

    Variation of Parameters - Higher order DE

    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...
  33. X

    Variation of parameters inhomogeneous DE help

    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...
  34. D

    How can I use variation of parameters to solve this differential equation?

    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...
  35. U

    Diff Eq: Variation of Parameters for 3rd-ODE's

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

    Second Order ODE - Variation of Parameters

    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...
  37. djeitnstine

    Variation of parameters method

    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) =...
  38. S

    Method of Variation of parameters

    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?
  39. U

    Trying to use variation of parameters

    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...
  40. A

    What is Variation of Parameters ?

    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...
  41. A

    Superposition and variation of parameters

    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...
  42. X

    Variation of Parameters Question

    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...
  43. N

    Differential Equations - Variation of Parameters problem

    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...
  44. H

    How Do You Solve a Fourth Order Differential Equation with Sinusoidal Forcing?

    [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...
  45. Saladsamurai

    Variation of Parameters on a system of Differential Eqs (Simple question)

    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...
  46. Saladsamurai

    Variation of Parameters (Integral Trouble)

    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...
  47. C

    How to Obtain Fundamental Solutions for Non-Constant Coefficient Equations?

    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...
  48. K

    Variation of parameters for higher order linear eq

    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...
  49. S

    A restriction within Variation of Parameters

    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...
  50. S

    Undetermined coefficients vs. Variation of Parameters

    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...
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