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.
TL;DR Summary: Variation of parameter VS Undetermined Coefficients
Hi all,
Suppose we want to solve the following ODE
2y''+y'-y=x+7
with two different methods: undetermined coefficients and variation of parameters.
The solutions to the homogeneous problem are given by y_1(x)=exp(-x) and...
Hi PF!
I am trying to solve an ODE by casting it as an operator problem, say ##K[y(x)] = \lambda M[y(x)]##, where ##y## is a trial function, ##x## is the independent variable, ##\lambda## is the eigenvalue, and ##K,M## are linear differential operators. For this particular problem, it's easier...
Homework Statement
Hi there, I don't nee help with solving a question, so much as understanding a step in the provided worked solution. It's using variation of parameters to solve the ode y''+ y = g(t). I've attached the steps in the picture file, and the bit after the word 'Now' what are they...
Homework Statement
The question I am working on is the one in the file attached.
Homework Equations
y = u1y1 + u2y2 :
u1'y1 + u2'y2 = 0
u1'y1' + u2'y2' = g(t)
The Attempt at a Solution
I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...
Homework Statement
Solve the DE by variation of parameters:
y'' - y = cosh(x)
Homework EquationsThe Attempt at a Solution
I got m = 1 and m = -1 so
y = c_1e^x + c_2e^{-x} + y_p
y_p = u_1e^x + u_2e^{-x}
The wonksian gave me -2
so
u_1' = \frac{\begin{vmatrix}
0 & e^{-x} \\
cosh(x)...
Hi PF!
Given operator ##B## defined as $$ B[u(s)] = c u(s) - u''(s) - \frac{1}{2 s_0}\int_{-s_0}^{s_0}(c u(s) - u''(s))\, ds$$ I'm trying to find it's inverse operator ##B^{-1}##. The journal I'm reading states ##B^{-1}## is an integral operator $$B^{-1}(u(s)) =...
Homework Statement
Determine the y_particular solution
Homework Equations
The Attempt at a Solution
I've tried this for hours but still don't get the correct value.
This is what I get:
The question is the same as the one I found from...
Hi everyone,
I am new to observations and observational terms!
I am reading the paper "constraining the time variation of the fine-structure constant" by Srianand et. al
in the section "constraining alpha with QSO absorption lines" there is a sentence saying "... rest wavelengths of MG II ...
$\tiny{242.17.8}$
2000
$\textrm{Solve the given equation by variation of parameters.}$
\begin{align*}\displaystyle
y''-10y'+25y&=2e^{5x}\\
\end{align*}
$\textrm{the homogeneous equation:}$
\begin{align*}\displaystyle
x^2-10x+25&=0\\
(x-5)^2&=0\\
x&=5\\...
Homework Statement
"By choosing the lower limit of integration in Eq. (28) in the text as the initial point ##t_0##, show that ##Y(t)## becomes
##Y(t)=\int_{t_0}^t(\frac{y_1(s)y_2(t)-y_t(t)y_2(s)}{y_1(s)y_2'(s)-y_1'(s)y_2(s)})g(s)ds##
Show that ##Y(t)## is a solution of the initial value...
Homework Statement
y''-4y'+4y=(12e^2x)/(x^4)
I am trying to solve this differential equation. I know you would use the variation of parameters method, and I am trouble with the wronskian. My solution manual doesn't actually use a wronskian so I can't verify my work
Homework EquationsThe...
Homework Statement
Use the method of variation of parameters to find a particular solution
Homework Equations
https://flic.kr/p/NqhtyQ
The Attempt at a Solution
https://flic.kr/p/NicCbN
[/B]
Can some find my mistake? The answer is just suplosed to be - 2/3te^-t[/B]
Homework Statement
Solve for the solution of the differential equation and use the method of variation of parameters.
x`` - x = (e^t) + t
Homework Equations
[/B]
W= (y2`y1)-(y2y1`)
v1 = integral of ( g(t) (y1) ) / W
v2 = integral of ( g(t) (y2) ) / W
The Attempt at a Solution
[/B]
yc= c1...
Kreyszig Advanced Engineering Mathematics shows the variation of parameter method for a system of first order ODE: \underline{y}' = \underline{A}(x)\underline{y} + \underline{g}(x) The particular solution is: \underline{y}_p = \underline{Y}(x)\underline{u}(x) where \underline{Y}(x) is the...
i was trying to get a particular solution of a 3rd order ODE using the variation of parameters method
the homogeneous solution is yh = c1 e-x + c2 ex + c3 e2x
the particular solution is yp=y1u1+y2u2+y3u3
as u1=∫ (w1 g(x) /w) dx , u2=∫ (w2 g(x) /w) dx , u3=∫ (w3 g(x) /w) dx
w =...
Just a question about the theory of solutions to differential equations?
Given a second order differential equation and two particular solutions y1 and y2, what is the best way to find the general solution?
i.e variation of parameters or something else
Use the variation of parameters method to find a general solution of $x^{2}y''+xy'-9y=48x^{5}$
$m^{2}-9=0$
$(m+3)(m-3)=0$
$m=3,-3$
$y_{h}=c_{1}x^{-3}+c_{2}x^{3}$
$W=6/x$ Don't really know how to do wronskian with latex so i didnt include the steps. But i need help with the rest of this. i...
Homework Statement
OK, this differential equation was technically created by me, because i need to clear my doubts.
Y'' + sqrt(X)*Y' + X^3*Y=3sin(x)
and actually just any initial conditions as long as the solution is something i can understand, let me expand my doubt further.
I've never solved...
Homework Statement
Find the general solution of the following equation:
u(t): u' = u/t + 2t
Homework Equations
y' + p(x)y = Q(x)....(1)
yeI = ∫ dx eIQ(x) + constant.....(2)
The Attempt at a Solution
I rearranged the equation to give:
u' - u/t = 2t
Then I considered the following...
Given a ODE like this:
y''(t) - (a + b) y'(t) + (a b) y(t) = x(t)
The general solution is: y(t) = A exp(a t) + B exp(b t) + u(t) exp(a t) + v(t) exp(b t)
So, for determine u(t) and v(t), is used the method of variation of parameters:
\begin{bmatrix}
u'(t)\\
v'(t)\\
\end{bmatrix}
=...
Can someone verify that my answer is correct ? Thanks in advance.
Use Variation of Parameters to find a particular solution to $y'' - y = e^t$
Solution:
$y_p = \frac{1}{2}te^t - \frac{1}{4} e^t$
I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order inhomogeneous ODEs), if possible.
The first is that, given the complementary solution...
Homework Statement
Solve by variation of parameters:
y" + 3y' + 2y = sinex
Homework Equations
Finding the complimentary yields:
yc = c1e-x + c2e-2x
The Attempt at a Solution
I set up the Wronskians and got:
μ1 = ∫e-2xsin(ex)dx
μ2 = -∫e-xsin(ex)dx
The problem is that I have no idea how to...
Homework Statement
The problem is attached as TheProblemAndSolution.png, and everything is typewritten, so it should be easily legible (but you will likely need to zoom into read the text since the image's height is significantly larger than its width).
Homework Equations
Differential...
Homework Statement
solve 4y''-4y'+y=16et/2
Homework Equations
v1= -∫ y2g/w
v2= ∫ y1g/w
The Attempt at a Solution
http://imgur.com/gxXlfdH
the correct answer is 2t^2 e^(t/2) instead of what i have though, i am not sure what i am doing wrong?
y'' + y = f(x) -- Variation of Parameters?
Homework Statement
Use variation of parameters to solve
##y'' + y = f(x), y(0) = y'(0) = 0.##Homework Equations
A description of the method is here: http://en.wikipedia.org/wiki/Variation_of_parametersThe Attempt at a Solution
The complementary...
I was given the problem,
"Find the general solution of the given differential equation."
\(y^{\prime\prime}+9y=9\sec^2(3t)\)
My work as follows, please let me know if this is correct and where to go from here. I have hit a roadblock of sorts.
\(y^{\prime\prime}+9y=9\sec^2(3t)\)...
I am working on the following problem:
Can someone please show or explain the steps to invert the phi matrix? I've given it a few tries, but I can't reach what the book has for the answer.
Please help! Thanks
Hi everyone, :)
One of my friends gave me the following question. I am posting the question and the answer here so that he could check his work.
Question:
This question concerns the differential equation,
\[x\frac{d^{2}y}{dx^2}-(x+1)\frac{dy}{dx}+y=x^2\]
and the associated homogeneous...
The ODE to solve via variation of parameters is ##(1-x)y''+xy'-y=(1-x)^2##.
Knowing that ##e^x## and ##x## are solutions to the homogeneous ODE.
Now if I call ##y_1=x## and ##y_2=e^x##, the Wronskian is ##W(y_1,y_2)=e^{x}(x-1)##.
According to...
Homework Statement
I must solve ##y''+2y'+2y=e^{-t}\sin t##.
I know variation of parameters might not be the fastest/better way to solve this problem but I wanted to practice it as I never, ever, could solve a DE with it. (Still can't with this one). Though the method is supposed to work...
x²y"(x)-3xy'(x)+3y(x)=2(x^4)(e^x)
=>y"(x)-(3/x)y'(x)+(3/x²)y(x)=2x²e^x
i don't know how to approach this problem because the coefficients are not constant and i am used to being given y1 and y2
HELP!
Here's the proof that I read for method of variation of parameters-
https://www.physicsforums.com/attachment.php?attachmentid=52267&stc=1&d=1351081780
What I couldn't understand is that how could one simply assume that
u'1y1+u2'y2=0 and
u'1y'1+u2'y'2=g(x)
I just don't understand...
I was recently trying to prove the variation of parameters formula for an nth degree equation, and I have come up with a question about the assumptions made during the derivation.
During the derivation we assume that: u1'y1(k) + u2'y2(k) + . . . + un'yn(k) = 0
for k < n-1.
It leads to the...
Homework Statement
y''-2y'+y = \frac{e^x}{1+x^2}
Homework Equations
u_1 = -\int \frac{y_{2}g(x)}{W}dx
u_2 = \int \frac{y_{1}g(x)}{W}dx
g(x) = \frac{e^x}{1+x^2}
W is the wronskian of y1 and y2.
The Attempt at a Solution
The characteristic equation for the homogenous solution...
Homework Statement
y'' + y' = 4t
Homework Equations
Use Variation of parameters!
The Attempt at a Solution
So I get homo of: c1 + c2 e^-(t)
From there I get a Wronskian of
-e^(-t)
Then I get variations 2t^2 and -4e^t(t-1)
Then get the answer of 2t^2 + 4t - 4
Btu...
Homework Statement
Find the particular solution to t^2 y'' - t(t + 2)y' + (t+2)y = 2t^3 given that y1 = t and y2 = tet are solutions. Also, require that t > 0
The Attempt at a Solution
Rewrite the original equation as y'' - ((t + 2)/t)y' + ((t+2)/t^2)y = 2t
So first I calculate the...
Homework Statement
Find the particular solution to t^2 y'' - t(t + 2)y' + (t+2)y = 2t^3 given that y1 = t and y2 = tet are solutions. Also, require that t > 0
The Attempt at a Solution
Rewrite the original equation as y'' - ((t + 2)/t)y' + ((t+2)/t^2)y = 2t
So first I calculate the...
Homework Statement
Solve y''+25y=10sec(5t)
Homework Equations
NA
The Attempt at a Solution
I believe I have the correct answer for yp which is:
2/5log(cos(5t))cos(5t)+2tsin(5t)
When I plug this into the Webwork field, it says it is incorrect. I checked my answer against...
Hey ya'll!
This is the equation under discussion:
y'' - 2y' - 3y = x + 2
I'm asked to use the method of variation of parameters to determine a solution for this differential equation, but I reach a point where my the equations just look too ridiculous to continue.
The point I have in...
Homework Statement
what is general solution of 2y'' - 3y' + y = ((t^2) + 1)e^tHomework Equations
my particular solution is: (e^t) ((2/3)(t^3) + 6t -4))
prof particular solution is: ((1/3)(t^3)(e^t)) - 2(t^3)(e^t) + 9(te^t)
The Attempt at a Solution
here is how i solved , i hope this is ok to...
Homework Statement
Solve using variation of parameters
y''' - 2y'' - y' + 2y = exp(4t)
Homework Equations
Solve using variation of parameters
The Attempt at a Solution
I got the homogenous solutions to be 1, -1, and 2.
So, y = Aexp(t) + Bexp(-t) + Cexp(2t) + g(t)
I got...
The question I'm trying to solve is:
y" - 6y' + 9y = \frac{exp(3x)}{(1+x)}
I formulated the Gen solution which are:
y1(x) = exp(3x) and y2(x) = xexp(3x)
I've then calculated the wronskian to get: exp(6x)
I then went onto to use the variation of parameters formula, which is where...
I just realized you can use variation of parameters (VOP) to solve for homogeneous 2nd order equations. I see it takes much longer to do so. But I was wondering why, if you use VOP, the u and v functions are 0. Is this because the coefficients of the homogeneous equation are constant, or...
Find the complementary solution of y^\left(4\right) + 2y'' + y = sint
Homogeneous Form would be y^\left(4\right) + 2y'' + y = 0
r^4 + 2r^2 + r = 0 \rightarrow r(r^3 + 2r + 1) = 0
This is where I'm stuck. Once I find y_c(t) I should be able to finish the problem, but I'm having trouble at this...
Homework Statement
I must solve (1-x)y''+xy'-y=(1-x)^2 knowing that y=x is a solution if the right hand side is 0. I must use this fact in order to obtain the general solution to the DE
Homework Equations
Variation of parameters?
The Attempt at a Solution
I'm looking at...
I've picked up a bit more since my last problem. I need to solve the following DE:
x^{2}\frac{dy}{dx}+x(x+2)y=e^{x}
I decided to use variation of parameters, so I re-arranged it like so:
\frac{dy}{dx}=\frac{e^{x}}{x^{2}}-(1+\frac{2}{x})y
Then solved the homogenous DE...
I need to find a solution to the following problem:
(x^{2}-1)\frac{dy}{dx}+2y=(x+1)^{2}
y(0)=0
I decided to try using variation of parameters. My teacher was unable to show any examples, and I'm having issues understanding the textbook.
From what I see I need to get it onto this form...
Homework Statement
Using the variation of parameters method, find the general solution of
x^{2}y" - 4xy' + 6y= x^{4}sin(x)
Homework Equations
y_{P}=v_{1}(x)y_{1}(x) + v_{2}(x)y_{2}(x)
v_{1}(x)'y_{1}(x) + v_{2}'(x)y_{2}(x)=0
v_{1}(x)'y_{1}(x)' + v_{2}'(x)y_{2}(x)'=x^{4}sin(x)...