Homework Statement
Solve:
(x^2-1)y'' + 4xy' + 2y = 6x, given that y_1=\frac{1}{x-1} and y_2=\frac{1}{x+1}.
Homework Equations
The Attempt at a Solution
Since both solutions are given, the solution to the homogenous system is:
y_h=C_1\frac{1}{x-1} + C_2\frac{1}{x+1}
And the...
Homework Statement
Find the general solution of the following homogeneous differential equations:
(i) \frac{du}{dx} = \frac{4u-2x}{u+x}
(ii) \frac{du}{dx} = \frac{xu+u^{2}}{x^{2}}
(You may express your solution as a function of u and x together)
Homework Equations
There are no...
I was wondering if some-one could give me some advice on how to complete the following equation:
\frac{du}{dx} = \frac{4u-2x}{u+x}
Whenever I try to complete it I get a function which I don't think should be integratible. Could some light be shed onto this matter?
I am having trouble getting to a solution for this differential equation
2(x^2+2x)y' - y(x+1) = x^2+2x -------- 1
for a series solution, we have to assume y = \sum a_{n}x^n ---------- 2
if we divide equation 1 by x^2 + 2x , we get (x+1)/(x^2+2x) for the y term, which is where my problem...
Hi all,
This is a problem I've been working on, off and on, for a few months now. It seems like it should be possible but I just can't figure it out.
Suppose you have a first order homogeneous function f(x_1, x_2, x_3). In other words, f has the property that: \lambda f(x_1, x_2, x_3)...
Homework Statement
\frac{1}{xy} \frac{dy}{dx} = \frac{1}{(x^2 + 3y^2)}
Homework Equations
used the substitutions:
v = \frac{x}{y} ,and
\frac{dy}{dx} = v + x \frac{dv}{dx}
The Attempt at a Solution
took out a factor of xy on the denominator of the term on the right hand...
in the following question i am asked to whow that Y1 and Y2 are basic solutions to the homogeneous equations
for 4.1)
y=ex=y' =y''
xy'' - (x+1)y' + y = x2
xex - (x+1)ex + ex = 0
none of these seem to work, what am i doing wrong?
Homework Statement
A function f is called homogeneous of degree s if it satisfies the equation
f(x1, x2, x3,... xn)=t^s*f(x1, x2, x3,... xn) for all t
Prove that the \sum from i=1 to n of xi * df/dxi (x1, x2, x3,... xn) = sf(x1, x2, x3,... xn).
Homework Equations
The Attempt at a Solution...
Homework Statement
For any a \in \left( -1,1 \right) construct a homeomorphism f_a: \left( -1,1 \right) \longrightarrow \left( -1,1 \right) such that f_a\left( a \right) = 0 . Deduce that \left( -1,1 \right) is homogeneous.Homework Equations
Definition of a homogeneous topological...
Homework Statement
Find the general solution to y'' + y = sec3(x)
The Attempt at a Solution
Well I can get the characteristic equation:
r2 + 1 = 0
r = +-i
Then the homogeneous solution is yh = C1excos(x) + C2exsin(x)
And I know y = yh + yp
but how do I get yp? I've never...
Hello,
In a book I'm reading about linear algebra it's mentioned that in order for the homogeneous system
Ax = 0
to have a solution (other than the trivial solution) the coefficient Matrix must be singular.
The thing is, I can't remember (the wikipedia page on homogeneous systems didn't turn up...
please help me :(
Electron inserted in a homogeneous electric field to measure 300 N / C, which
directed vertically upwards. The initial velocity of the electron is far
5,00 10^ × 6 m/s and goes to 30 degrees, above the skyline. a) Find the maximum height that
reaches the electron above the...
Homework Statement
Find y as a function of x if
y'''−11y''+28y'=0 y(0)=1 y'(0)=7 y''(0)=2
I have one attempt left on this question. Could someone verify my answer for me?
Homework Equations
The Attempt at a Solution
(use t as lamda)
t^3-11t^2+28t=0...
Homework Statement
Solve 354y`` −692y` + 235y =0
y(0) = 7
y`(0) = 4
Homework Equations
The Attempt at a Solution
First I divided the equation by 354 to get y`` - 1.56y` + 0.894y = 0.
Then I found the roots of this to be 0.94, repeated twice.
For repeated roots the solution looks like y=...
Hello,
how would you solve an homogeneous system of the form A\mathbf{x}=0, with the constrain <\mathbf{x},\mathbf{x}>=1. The matrix A is symmetric, but I don't know if it matters.
There should be a method involving eigenvalues, but strangely enough, I can't find it in any book.
Thanks!
Homework Statement
If a linear homogeneous system Ax=0 has a non - trivial solution and A is an n x n matrix, then (choose ALL correct answers)
A. A has rank less than n
B. Each system Ax=b with the same coefficient matrix A has a solution
C. A is row equivalent to I
D. If Ax=b has...
Homogeneous Linear DE's -- solving IVP's
Homework Statement
Solve the given IVP:
d^2y/dt^2 - 4 dy/dt -5y = 0; y(1)=0, y'(1)=2
Homework Equations
N/A
The Attempt at a Solution
I've solved and got the general solution y=c1e5t+c2e-t
I'm plugging in the following to solve...
Homework Statement
(4y4-9x2y2-144)dx - (5xy3)dy = 0
Homework Equations
substitute y = xv
dy = dx v + dv x
The Attempt at a Solution
after substituting i got
(4x4v4-9v2x4-14x4)dx - (5v3x4)dx.v + dv.x
= (4v4-9v2-14)dx - 5v3(dx.v + dv.x) = 0
= dx(4v4-9v2-14-5v4)+dv(-5v3x)= 0...
Homework Statement
Claim:
The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space".
Proof:
Assume Lu=0 and Lv=0 (i.e. have two solutions)
(i) By linearity, L(u+v)=Lu+Lv=0
(ii) By linearity, L(au)=a(Lu)=(a)(0)=0
=> any linear...
Homework Statement
Given a differential equation.
ie. y'' +y' + 1 =0 (THIS IS NOT THE PROBLEM THAT I AM SOLVING)
Homework Equations
No equations
The Attempt at a Solution
The equation above is not what I'm working with, but an example of a differential equation problem that I was...
Hi guys,
I've been out of the loop with differentials for some time, and was hoping to get some direction with these..
Given that:
(x^2)y" + 2x(x-1)y' - 2(x-1)y = 0
A) Explain why the general theory doesn't guarantee a unique soln to the equation satisfying the initial conditions y(0)...
Homework Statement
Prove that in the homogeneous deformation, particles which after the deformation lie on the surface of a shere of radius b originally lay on the surface of an ellipsoid.
Homework Equations
homogeneous deformations are motions of the form:
xi=ci + AiRXR
where ci...
Hi
I'v got a maths exam on Tuesday for my 2nd year of chemial engineering.
Been going through a past paper and have been going over 2nd order homogenous DE's
Im at the stage of calculating the roots (wether repeated or 2 distinct roots) I take the easy path like so:
E.g m2 + 4m + 4 =...
Can someone explain to to me how to find the general solution of the fourth order ODE
y''''-y''=0
Right now I have
y(x)=a+b*x+c*e^-x+d*e^x
where a,b,c and d are constants.
Not sure if this is correct just wanted to double check.
Homework Statement
4xy'' + 2y' + y = 0
2. The attempt at a solution
In class, we were given that y1 = c1Cos(\sqrt{}x). We then used reduction of order to figure out the other solution
Yet, I've been trying to figure out, is how do you get y1 in the first place? To me, it doesn't...
Homework Statement
y2dx -x(2x+3y)dy =0 I have to recognize the equation and solve it
Homework Equations
The Attempt at a Solution
I did y2dx - (2x2+ 3yx) dy=0
which is a homogeneous now
after I substitude x=uy
dx=udy + ydu
I stuck here after the substitution...
Hey guys, I've been stuck on this problem for a good hour... I have no idea how to finish it up.
Homework Statement
Solve: dy/dx = 4y-3x / (2x-y)
Use homogeneous equations method.
Homework Equations
Answer : |y - x| = c|y + 3x|5 (also y = -3x)The Attempt at a Solution
dy/dx = 4y-3x / (2x-y)...
Homework Statement
So the problem goes like this:
dy/dx = ( x^2 + xy + y^2 ) / x^2
a) Show that it is a homogeneous equation.
b) Let v = y/x and express the eqn in x and v
c) Solve for y
Homework Equations
(Included)
The Attempt at a Solution
a) dy/dx = ( x^2 + xy + y^2 ) / x^2 * [(1/xy)...
I read that the 'Homogeneous Differential Equation' is one which has form \frac{\mathrm{d}y}{\mathrm{d}x} = f\left(\frac{y}{x}\right) but I came across one example which was \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x+y}{x+5y} and said that is 'Homogeneous Differential Equation' Too which has 'x'...
Homework Statement
Please help me with this, I would really appreciate it:
Using the fact that y1 = x-1/2 cos x is a solution of the associated homogeneous problem, obtain the general solution of
x2 y'' + x y' + (x2 - 1/4)y = x3/2
The Attempt at a Solution
Well the first thing i...
Homework Statement
Find the general solution of \frac{dy}{dt} + 2y = 3t^2 + 2t -1
Homework Equations
The Attempt at a Solution
So just worrying about the right side
y_p = at^2 + bt + c
so \frac{dy_p}{dt} + y_p = 2at + b +at^2 + bt + c = 3t^2+2t - 1
at^2 = 3t^2...
Homework Statement
Can y = sin(t^2) be a solution on an interval containing t = 0 of an equation y'' + p(t)y' + q(t)y = 0 with continuous coefficients?
Homework Equations
The Attempt at a Solution
y = sin(t^2)
y' = 2tcos(t^2)
y'' = 2cos(t^2) - 4t^2sin(t^2)
2cos(t^2) -...
First Order Linear Non-Homogeneous Equation
Homework Statement
I need to solve for e(t)
Homework Equations
Do I use Laplace Transform for the last integral?
The Attempt at a Solution
\begin{subequations}
\begin{eqnarray}
\nonumber
\dot{\hat{{\cal E}}}(t) &=& -\kappa...
I would very much appreciate if anyone can help me with this problem; I have approached it from many different angles to no avail.
The position x(t) of a particle moving along the x-axis is governed by the differential
equation:
x'' + kx' + (n^2)x = 0 , and initially x(0) = a, x'(0) = u...
I recently read an article about whether the http://dailyphysics.com/" and/or isotropic. (the story is at the top - sorry I couldn't get the link to the permanent article to work here) “gargantuan ripples in the density of matter across the universe, known as baryon acoustic oscillations” is...
Differential Equation w/ Homogeneous Coefficients - y=ux substitution
I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class.
Homework Statement
(x+y){dx} - (x-y){dy} = 0
Homework Equations
y=ux, {dy} = u{dx} + x{du}
The...
Hello everyone!
I'm trying to solve 2 questions for my assignment on homogeneous ODEs
I can solve ODE's with variation of parameters and with the method of undetermined coefficients, but these 2 methods seem useless when the coefficients are not constant :
(x^2)y" - 2xy' -54y = 0
and...
Homework Statement
Let A * x= 0 be a homogeneous system of n linear equations in n unknowns, that has only the trivial solution. Show that if k is any postive integer, than the system A^k * X = 0 also has only the trivial solution.
The Attempt at a Solution
I'm so lost please help...
Homework Statement
Determine the x-coordinate of the mass center of the homogeneous hemisphere with the smaller hemispherical portion removed?
I know what the answer should be it's Xcm = 45/112 R
Homework Equations
The center of mass R of a system of particles is defined as the average...
Homework Statement
"Solve the following ODE's:"
"3u+(u+x)u'=0"
This is our first weeks homework and he went this through this so quickly in class.
Homework Equations
None. x and u are both variables.
The Attempt at a Solution
I know it is homogeneous and non linear. I tried v-substitution...
Homework Statement
The following DEs are homogeneous and can be solved using a substitution y=ux or v=xy. I can solve them, I'm just don't see how they are homogeneous.
How is it that with there being an exponent (y/x) that this is homogeneous. I don't see how I could pull out t\alpha...
The concept of invariance of an object, property, etc... should always expresses with respect to something else:
time-invariant means static.
space-invariant means homogeneous.
direction invariant means isotropic.
"Something" can have just one of those types of invariance, all three of...
this is the given equation
y'=(4y-3x)/(2x-y)
and here is all the work I've done so far:
(4v-3)/(2-v)=v+x*dv/dx
i moved v over and came up with this
(-3+2v+v^2)/(2-v)=x*dv/dx
did a flip
(2-v)/(-3+2v+v^2)dv=dx/x
by partial fractions I got a=-3/2 and b=1/2
so...
Homework Statement
Given,
(y+2)dx + y(x+4)dy = 0, y(-3) = -1Homework Equations
v=y/xThe Attempt at a Solution
I've been REALLY struggling with homogeneous equations for some reason...I just don't understand them all.
so far I've tried two things.
(1)dx -(y)dy
----- -------
(x+4)...
Hi everyone. I am really confused at the moment learning about Second Order homogeneous linear differential equations. I lay out the background of what I would like to understand. So I understand the actual maths that goes into the diff's, but I do not understand why it should be so, given the...
Hey all, i think I'm doing most of this right, but I'm missing a coefficient somewhere when integrating or something...
Homework Statement
Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation...
I have a question:
If the vectors of v & u are solutions of the nonhomogeneous linear system Ax=b, then ru+sv is a solution of the nonhomogeneous system for any real values of r and s.
is this statement true?
is this statement true for homogeneous systems too?