What is Particular solution: Definition and 104 Discussions
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
If I have been given a system of inhomogeneous linear ODEs,
$$
\vec{x'} =
\begin{bmatrix}
4 & -1 \\
5 & -2 \\
\end{bmatrix}
\vec{x}
+
\begin{bmatrix}
18e^{2t} \\
30e^{2t}\\
\end{bmatrix}
$$
I have found its particular solution to be:
$$
1/4
\begin{bmatrix}
-31e^{2t} - 25e^{6t} \\
85e^{2t} -...
I identified the root 1 with multiplicity 1 and the root 2 with multiplicity 1. So The characteristic equation is ((m-1)^2)*(m-2)=0. Simplifying and substituting with y I found: y'''-4y''+5y'-2y=0.
So now I've realized that this is actually describing y(t)=(C1)*(e^t)+(C2)*(e^t)+(C3)*(e^2t) and...
Homework Statement
I am trying to solve 2nd order in-homogeneous equation of y(x) (given below). I was able to get the Homogeneous solution , but i am not able to create the Ansatz for the particular solution.
It would be really helpful if anyone suggests an Ansatz for this equation and also...
Homework Statement
I'm fine with the first part. Part b) is causing me trouble
http://imgur.com/xA9CG5G
Homework EquationsThe Attempt at a Solution
I tried subbing in the solution y1 into the given equation, but I'm not sure how to differentiate this, i thought of using integration by parts...
Homework Statement
In question #1(b)'s answer, should x_1 = 4 (instead of -4)?
Homework Equations
PDF of problem #1(b):
https://www.dawsoncollege.qc.ca/public/programs/disciplines/math/exams/201-nyc-05-computer_science-winter2011.pdf
Reduced row echelon form of the matrix I obtained from the...
Homework Statement
Consider two masses m connected to each other and two walls by three springs with spring constant k. The left mass is subject to a driving force ## F_d\cos(2 \omega t) ## and the right to ## 2F_d\cos(2 \omega t) ##
Homework Equations
Writing out the coupled equations:
$$...
ASSUMPTIONS:
BIBO/stable systems
NOTE: zero here does not mean the roots of the denominator in a transfer functionTRUE/FALSE -Please provide feedback- some answers are based on ODE example listed below
1/True) The Homogenous Solution is either zero or transient.; i.e. it can never be steady...
CONCEPTUAL QUESTIONS:
-Does the Homogenous Solution represent the Transient Response?
Let me specify. For a N-DOF spring, mass, and damper mechanical system:
-Does the Homogenous Solution represent the Transient Response for given mechanical system?
MY ANSWER:
Yes.
ASSUMPTIONS:
-only...
Homework Statement
Find a particular solution of y"-5y'+6y=-e^(x)[(4+6x-x^2)cosx-(2-4x+3x^2)sinx].
Homework Equations
None.
The Attempt at a Solution
yp=e^(x)[(Ax^2+Bx+C)(cosx-sinx)]
y'p=e^(x)[(Ax^2+Bx+C)(-sinx-cosx)+(2Ax+B)(cosx-sinx)]+e^(x)[(Ax^2+Bx+C)(cosx-sinx)]...
Homework Statement
Find a particular solution of y"+2y'+y=8x^2*cosx-4xsinx. The answer is yp=-(14-10x)cosx-(2+8x-4x^2)sinx.
Homework Equations
None.
The Attempt at a Solution
r^2+2r+1=0
(r+1)^2=0
r=-1, -1
y=Axe^(-x)+Be^(-x)
So what should the initial guess yp be for this problem? How to find...
Homework Statement
Particular solution of
y" - y' - 2y = e^(2x)
Homework Equations
None
The Attempt at a Solution
This makes no sense to me, why do I have to use the solution of the form
y(t) = cxe^(2x)
For the problem above, but when I switch the signs and it becomes
y" - y' + 2y =...
Homework Statement
dy/dx = (9x²)/(9y²-11) , y(1)=0
Homework EquationsThe Attempt at a Solution
I've found the particular solution:
3y³-11y = 3x³-3
The solution is defined for -⅓√11<y<⅓√11,
What about the x values? Plugging the y values into the equation doesn't work of course..
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
imgur link: http://i.imgur.com/8TOXi9t.png
I am comfortable with the need to multiply the polynomial in front of e^{2x} by x^3, that makes perfect sense in terms of what the text has already said about how no term in the particular solution should duplicate a term in the complementary solution...
Given $y'' + 3y'-4y= sin \omega t $, I used an ansatz of $y_p = A sin \omega t + B cos \omega t$
$\therefore y' = A \omega cos \omega t -B \omega sin \omega t, y'' = -A \omega^2 sin \omega t - B \omega^2 cos \omega t $
Substituting and equating coefficients, I get $ -A \omega^2 - 3B \omega -...
Homework Statement
Find a particular solution to
##y'' - 3y' + 2y = 6x^2##
I don't understand how/why the value of c has been determined. I'm hoping it is a mistake in the solution, but knowing me, it's probably my mistake.
Homework EquationsThe Attempt at a Solution
assume a solution of the...
Hey! :o
Suppose we have a non-homogeneous differential equation $Ly=f$ in the ring of exponential sums $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$, then $f=\alpha_ie^{b_i x}$, right?
When $b_i$ is a root of the characteristic equation of the homogeneous equation of multiplicity...
Mentor note: moved from mathematics forums, therefore no homework template
Hello mates. I have a System of Differential Equations on my hands.
$$ W' = \left(\begin{matrix} 4 & 2 \\ 2 & 4 \end{matrix} \right) W + \begin{pmatrix} 6 \\6 \end{pmatrix}$$
where transposed W is (x y z). I have found...
Homework Statement
I started learning about solving non homogeneous linear differential equations in class and I am a bit clueless on how to solve them since I've never had a prior experience with much of differential equation.
I am trying to find the particular solutions to the equation...
Hey! :o
When we have the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=f(x)$$ and the non-homogeneous term $f(x)$ is of the form $e^{mx}P_n(x)$ we know that the particular solution is $$y_p=x^k(A_0+A_1x+ \dots +A_nx^n)e^{mx}$$ where $k$ is the multiplicity of the eigenvalue...
Homework Statement
Find both a particular solution yp (via the method of educated guess) and a general solution y.
y(2)-6y(1)=25sin(6x)Homework EquationsThe Attempt at a Solution
This was my guess:
yp(x)=Asin(6x)
I then took derivatives, plugged them in, and simplified to get this...
how can I find the particular solution for something like this
x = 120 cos 6t
I know how to find a particular solution for equations like x''+x'+x=0, you can easily find the characteristic equation then find the particular solution, but I have no idea how to deal with an equation like that..
Homework Statement
The problem reads:
Find a homogeneous linear differential equation with constant coefficients that has the following particular solution:
yp = e^(-t) + 2te^(t) + t^(2)e^(t) - sin(3t)
Express your equation in differential operator form. (Hint: What annihilators would...
Homework Statement
Solve: A*sin(ωt + Θ) = L*i''(t) + R*i'(t) + (1/C)*i(t). Where: A=2, L = 1, R=4, 1/C = 3 and Θ=45°.
Homework Equations
The system has to be solved by i(t) = ih + ip. I gave the values to A, L, R, 1/C and Θ. I can also give values to ω, but I've come to a doubt when solving...
Hi,
I have a question about how to find the particular solutions when trying to solve recurrence relations. For example, trying to solve
an+2 = -4an + 8n2n ,
I begin with finding the roots in the characteristic polynomial associated with the homogeneous equation, so r1 = 2i and r2 = -2i...
Hello,
I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation.
It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by:
Ygeneral = Yhomogeneous + Yparticular
My question: Is it...
Given the nonhomogenous differential equation y'' + 3y' + 2y = -10e^(3t), the roots are r = -2 & -1, & the characteristic eq'n is yc(x) = c1e^(-2t) + c2e^(-t)
How do we go about setting up the particular solution?
There is no repetition between terms so I know that we do not add a variable...
Hi,
I'll give some background, say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0 and y = -d, respectively.
The structure has scalar potentials inside it as so:
As you can see the vector fields cancel out on one side, As it...
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:
$y_p = \frac{1}{2}te^t - \frac{1}{4} e^t$
Note that the general solution to $y'' - y = 0$ is $y_h = C_1e^t + C_2e^{-t}$
In the following, use the Method of Undetermined Coefficients to find a particular solution.
a)$y'' - y = t^2$
So here is what I have so far
$y_p = At^2 + Bt + C$
$(y_p)'' = 2A$
Ive got $A = -1, B = 0 , C = 0$
so...
I have an exam in 2 days and I am still getting confused on how to find yp , the particular solution.
for example
y''+ 4y = 9te^t+ 4
or tsin(2t) + 2
Is it only by guessing and that's it? I still can't answer these questions 100% correctly. I would like to have your advises to find the best...
Homework Statement
y'' + 9y = 3sin(3x) + 3 + e^{3x}
Homework EquationsThe Attempt at a Solution
This is my first post here so let me know if I've done anything wrong, I've been looking at questions here for a long time though ^^.
So the problem asks me to solve for one particular solution...
Homework Statement
Find the particular solution of the differential equation that satisfies the initial condition.
dx/dy = e^x+y x(1) = 0
Homework Equations
it does specifically state dx/dy not dy/dx
The Attempt at a Solution
dx/dy= e^x *e^y
1/e^x dx=e^y dy...
I am studying Introduction to linear Algebra by Gilbert Strang while calculating the particular solution P for $Ax=b$,he made the free variables $0$ to calculate the particular solution and said that P along with linear combinations of null space solutions make up the complete set.
I understood...
I know nothing about DEs, so this may be a silly question.
I'm given some time varying (x_t)_t and a constant r, and I want to solve the equation u_t = rx_t + \dot u_t for u.
What I know so far is that (solving the homogeneous equation) if \bar u is some particular solution, then any u is...
Find a particular solution for the following equation:
y"+2y'+y=12.5e-t
I'm not sure on which method to use. Here's my attempt using the undetermined coefficients method:
→y"+2y'+y=12.5e-t
r2+2r+1=0
r=-1 *not even sure if this part is useful
→yp=e-t
yp'=-e-t
yp"=e-t...
Homework Statement
Verify that the vector functions x_{1}=\begin{bmatrix}e^{t}\\ e^{t}\end{bmatrix} and x_{2}=\begin{bmatrix}e^{-t}\\ 3e^{-t}\end{bmatrix} are solutions to the homogeneous system
x'=Ax=\begin{bmatrix}2 & -1 \\ 3 & -2 \end{bmatrix} on (-\infty ,\infty )
and that
x_{p}...
Find the particular solution to Dy/Dx = 3x^2 + 1 when y(0) = 3
I've looked everywhere for steps to solve this problem, and every website I have been to has taught me how to do a question like this when each side has a Y (is separable) but I can't find out how to do one like the question above...
Homework Statement
Please see attachment
Homework Equations
Please see attachment
The Attempt at a Solution
Please see attachment. I am supposed to compare coefficients but it doesn't seem possible. what do I do next? (also, if i have posted this in the wrong forum...I...
Homework Statement
Could someone please help me do this problem?:
“Write the complete solution as x_p plus any multiple of s in the nullspace:
x + 3y + 3z = 1
2x + 6y + 9z = 5
–x – 3y + 3z = 5”
The answer is x = x_p + x_n = {{-2},{0},{1}} + x_2 {{-3, 1, 0}}.
Homework Equations
Ax...
difference between particular integral and particular solution..??
particular integral and particular solution are used to find the solution of differential equation..but sometimes they are used interchangeably but what's the difference between two..??
please state examples..!
Hi!
I performed the blue bottle experiment using Methylene Blue, water, NaOH and Glucose.
The reaction went fine but I have planned to make a gel of the reactants EXCLUDING glucose and then add glucose in controlled amounts to this gel to see the effects.
I know it may sound like an...
Homework Statement
For 1st order linear ordinary differential equation, how do "unique solution" and "particular solution" differ?
Homework Equations
if dy/dx = f(x,y) and partial f(x,y) with respect to y are both continuous, then there exists a unique solution within a region R...
Homework Statement
Find the appropiate particular solution form for
y''' - y'' - 12y' = x - 2x*e^-(3x)
DO NOT SOLVE FOR THE COEFFICIENTS
Homework Equations
Just guessing proper solution forms for nonhomogeneous second order D.Es with undetermined coefficients.
The Attempt at a Solution...
1. Homework Statement
The problem says:
For the next circuit obtain a particular solution for R3 when v1=v2=l1=l2=0 at t=0.
vs= cos(wt) dc source= 10v *R3 is the 1k resistor.
http://postimage.org/image/fxtu2blaf/
2. Homework Equations
Kirchoff current law...
There are two questions that I am trying to solve on web assignment. The goal is to find a general form of a particular solution to each ODE. The question asks me to represent all constants in the solution using "P,Q,R,S,T..etc.", in that order.
1. y'''-9y''+14y'=x2
2. y''-9y'+14y=x2e4x...