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. D

    I am pissed And Variation of parameters. who, what, when, where, why?

    Well, i really i am pissed at the moment because of this. I have been trying to figure out if there is a proof/justifaction for the method of variation of parameters and have had absolutly no luck. I have searched the internet and almost every differential equation books in my university library...
  2. K

    How can I determine the correct solutions for a variation of parameters problem?

    Homework Statement x^2y''-2xy'+2y=x^3cosx Find a general solution by using variation of parameters 2. The attempt at a solution I already solved this one, but I have 4 questions: 1. I found the solutions x and x^2 to the homogeneous equation by inspection. Is this the only way to...
  3. I

    Solving Differential Equations with Variation of Parameters

    Homework Statement Given that y=x^2 is a solution to the differential equation: (x^2)y'' + 2xy' - 6y = 0 <--- Eq.(1) find the general solution of the differential equation (x^2)y'' + 2xy' - 6y = 10(x^7) + 15(x^2) <--- Eq.(2) Hence write...
  4. I

    Solving ODE with Variation of Parameters

    Hi i need to use Variation of Parameters to solve this ODE x^2 y'' - 2xy' + 2y = x^(9/2) So far I was thinking to use Euler's Equation and I really don't know if it will work please help me out with a hint. THanks.
  5. K

    Nonhomogenous LODE (Method of Variation of Parameters)

    Nonhomogenous LODE (Higher Order) - Method of Variation of Parameters x^3y''' + x^2y'' - 2xy' + 2y = x^3log(x) y(1) = \frac{10}{32} y'(1) = -\frac{24}{32} y''(1) = -\frac{11}{16} I know that \inline y = y_h + y_p and that I probably should use the method of variation...
  6. W

    #2 Variation of parameters with complemetry EQ (Diffy Q)

    Given that http://forums.cramster.com/Answer-Board/Image/cramster-equation-200641003458632802260985487500893.gif is the complementary function for the differential equation http://forums.cramster.com/Answer-Board/Image/cramster-equation-200641003635632802261951581250765.gif use the variation...
  7. W

    Variation of Parameters Differential Eq.

    Use the method of variation of parameters to find a particular solution of http://forums.cramster.com/Answer-Board/Image/cramster-equation-2006491914366328020687673812501253.gif ok i know first i do...
  8. H

    Can Variation of Parameters Solve This First-Order Differential Equation?

    I'm not currently in a class, but I'm doing this for fun.. but technically I would still call it coursework, so I'm posting it here.. I'm studying Redheffer/Sokolnikoff's Mathmatics of Modern Engineering.. and I find a problem on page 75, use the method of variation of parameters to find the...
  9. M

    Variation of parameters problem, very BIG, but i think i'm right so far Diff EQ

    OKay everyone, this is a big f'ing problem (to me anyways) and its only worth 1 point! But I'm doing it anyways. So here was my attempt, everything seems to be working out like it should but look at what u1 came out too, what am i going to do with that mess? Also do u see any mistakes...
  10. A

    Solution by variation of parameters

    i'm confused about that method: 1) when proving that method works, why do you have to make u and v satisfy the 2nd condition u`y1+v`y2=0 2) when you're integrating to find yp, why do you leave out the constant that results from the integration?
  11. A

    Undetermined Coefficients / Variation of Parameters

    I know how to solve the following ODE with variation of parameters: y''+4y=4\sec{\left(2t\right)}. Is there any way to solve this with undetermined coefficients? So far I have tried Yp=Acos(2t)+Bsin(2t), but that didn't work. Thanks for the help.
  12. P

    Solve differential equation with variation of parameters

    solve using method of variation of parameters y''-y = 2/(1+e^x) y'' ==> second order
  13. E

    Understanding Variation of Parameters in Linear Differential Equations

    I have been trying to get this for a while and can't figure it out: if a fundamental matrix of the system x' = Ax is X(t) \left(\begin{array}{cc}e^t&0\\0&e^{-t}\end{array}\right) find a particular solution yp(t) of x' = Ax + [e^t, 2]^{transpose} such that yp(0) = 0 So, I got u(t) =...
  14. RadiationX

    Solving D.E. with Variation of Parameters Technique

    I need to solve this D.E. x^2y''-xy' + y = x^3 i'm supposed to use the variation of parameters technique. in that technique i need to get a coeffecient of 1 in the first postion of y'' and then sove the homogenous D.E. y''-\frac{y'}{x} +\frac{y}{x^2}=0 the above leads to...
  15. C

    What are the steps for finding u1 and u2 in the variation of parameters method?

    Hello, I'm trying to understand this concept. Jere's the problem I'm doing. I have to find the general solution for: y'' + 36y = -4xsin(6x) So you then solve for your characteristic equation and get lamda = +/- 6 so y1 = e^-6x and y2 = e^6x You get your matrix for w, w1, and w2. w =...
  16. T

    Problem using variation of parameters

    the problem is fin the general solution of the differential eq : y''+y=2sect + 3 (-pi/2 < t < pi/2) using variation of parameters. I just needed a check to make sure my answer was correct. r^2+1 = 0 r= -i r= i y1= cost y2= sint g(t)= 2sect+ 3 y(t) = c1cost + c2sint + Y(t)...
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