I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...
I've found the general solution to be y(x) = C1cos(x) + C2sin(x).
I've also found a recursion relation for the equation to be:
An+2 = -An / (n+2)(n+1)
I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?
Any help would be...
In Arnold's book, ordinary differential equations 3rd. WHY Arnold say Tg:M→M instead of Tg:G→S(M) for transformations Tfg=Tf Tg,
Tg^-1=(Tg)^-1.
Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a...
hi everyone, i'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book...
1. The problem statement, all variables, and given/known data
Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for ##u(x,y)##. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations.
According to the solution...
I want to find solution to following ODE
$$ \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) $$
I have solved it with integrating factor method with ## I=\exp^{\int \frac{1}{D} \alpha^2 dt} ## as integrating factor and ##\frac{K}{S_s} = \frac{1}{D} ##
I have...
During solution of a PDE I came across following ODE
##
\frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t)
##
I have to solve this ODE which I have done using integrating factor using following steps
taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...
1. Homework Statement
I am having trouble proving if the equation i have found for number 1 is correct. I have posted my solution to get back to the main problem in the first photo below.
For number 2 I am having trouble isolating for 1 y(x). Did i do the integration and setup properly?
2...
1. Homework Statement
The equation given:
dy/dt = 3*y
A basis for the space of solutions is required.
3. The Attempt at a Solution
According to me it is e^(3*t) but it has turned out false. Why? I am considering the answer "The basis is the set of all functions of the form c*e^(3*t)...
I need to introduce Simple Harmonic Motion to a group of high school students studying physics. They don't know anything about differential equation except the method of separation of variables. Also, they have limited knowledge on complex numbers like eiωt. However, I don't want to just give...
1. Homework Statement
I'm actually a tutor, and a student of mine at uni has the following differential equation with initial conditions to solve
imgur link: http://i.imgur.com/ptuymQv.gif
From y(t) = c_1sin(3t) + c_2cos(3t), it is not possible to solve for the constants using the given...
1. Homework Statement
Solve the following equation.
2. Homework Equations
( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0
3. The Attempt at a Solution
M = ( 3xy4 + 2xy )
N = ( 2x3y3 - x2 )
∂M/∂y = 12x2y3 + 2x
∂N/∂x = 6x2y3 - 2x
Then this equation looks like that the integrating factor is...
1. Homework Statement
Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0##
Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##.
Then solve the differential equation with...
Given a linear homogeneous 2nd order ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ the general solution is of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{1}(x)$$ where ##c_{1},c_{2}## are arbitrary constants and ##y_{1}(x), y_{2}(x)## are linearly independent basis solutions.
How does one prove that...
1. Homework Statement
Find a curve that passes through point A(2,0) such that the triangle which is defined with a tangent at arbitrary point M, axis Oy and secant \overline{OM} is isosceles. \overline{OM} is the base side of a triangle.
2. The attempt at a solution
Function passes through...
1. Homework Statement
Find the Green's function $G(t,\tau)$ that satisfies
$$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$
under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$
Then...
Hello, I really need a good book on ordinary differential equations with applications on Quantum Mechanics, as I will be attending a course on QM but I do not have the proper mathematical background that is needed.
1. Homework Statement
Y''-((Y')^2)+(C1*exp(Y))=C2
C1 and C2 are constants.
exp = e
2. Homework Equations
No clue how to start this
3. The Attempt at a Solution
Y'=A=dY/dt
Y=At+C3 (not sure)
A'-(A^2)+C1exp(At+C3)-C2=0
A'-(A^2)+C1exp(C3)exp(At)=0
let C=C1*exp(C3)
A'-(A^2)+Cexp(At)=0
The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this
\begin{equation}
y''(t')+2y''(t)-2y(t)=0
\end{equation}
And the initial conditions are
\begin{equation}
y(0)=y'(0)=0, y''(0)=0
\end{equation}
The problem is with the first...
Im trying to implement the implicit Euler method in high-performance software for micromagnetic simulations, where I'm restricted to using the driving function of the ODE (Landau-Lifshitz equation) and the previous solution points. This obviously not a problem for an explicit method, since we...
Homework Statement
Well I am looking for the particular integral of:
d2y/dt2 + 4y = 5sin2t
The attempt at a solution
As f(t) = 5sin2t, the particular integral yPI should look like:
yPI = Acos2t + Bsin2t
dyPI/dt = -2Asin2t + 2Bcos2t
d2yPI/dt2 = -4Acos2t - 4Bsin2t
Subbing in to the differential...
i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform.
Laplace can be used to analyze unstable systems.
Fourier is a subset of laplace.
Some signals have fourier but laplace is not defined , for instance cosine or sine...
1. Homework Statement
I stumbled upon a problem and i can't establish the ODE to solve it, from there on i believe i can solve the ODEs if they have regular analytical solving methods (translated from Spanish, will sound a bit weird)
Car race, 2 pilots (a and b) participate in a drag race. They...