Tagged Questions
2
votes
3answers
72 views
General solution of differential equation of order 3
Please ,how to find that the general solution of $u'''(t)=e(t) , t\in [0,1]$ is given by
$u(t)=c_0+c_1t+c_2 t^2 +\frac12 \int_0^t (t-s)^2 e(s) ds$
$e:(0,1)\rightarrow \mathbb{R}$, and $e\in ...
2
votes
4answers
63 views
initial value problem: y'' + 4y = f(t) , y(0)= y'(0)=0. f(t) = { 0 if t <3; t if t >3}
Solve the initial value problem:
$$y'' + 4y = f(t) , y(0)= y'(0)=0. $$
where
$$ f(t) = \begin{cases} 0 &t < 3 \\ t & t > 3\end{cases} $$
I've solved for the homogeneous equation, $y'' ...
0
votes
1answer
38 views
Question about eigenvalues
I have this :
i dont understand why they write $\lambda=m^2 , m\in \mathbb{N}\cup\lbrace0\rbrace$ ,
it's right that $\lambda=m^2$ is the eigenvalues of $(P_0)$ ,but $0$ is not an eigenvalue !.
...
0
votes
1answer
37 views
Linear Differentiation
I have to determine whether there is normal linear differentiation equation $a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ on $\mathbb{R}$ such that $u_1, u_2 \in C^2(\mathbb{R})$ defined by $u_1(x) = x, u_2(x) ...
0
votes
1answer
37 views
Ordinary differential equations with double resonance
I want to know what is the definition of "resonance, double resonance" in
ordinary differential equations with double resonance
Please,
Thank you.
1
vote
1answer
29 views
Eigenvectors and differential equations
I was able to find part (a), and I got 4 and -1 for the eigenvalues and from these values I got eigenvectors of [1,1] and [-3,2], but I don't know what to do for part (b) and (c)
1
vote
1answer
30 views
Solving Euler-equation alike 2nd order DE with disturbing RHS
For a homework problem, I have to solve
$$ t^2 \ddot{x} - 3 t \dot{x} + 3x = t^2 $$
which seems quite similar to the Euler Equation, which I would know how to solve, apart from the disturbing $ t^2 ...
2
votes
2answers
30 views
Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
0
votes
1answer
26 views
Euler's method for second order differential equation
Not really homework but sample exam.
The question is to use Euler's Method to approximate Y:
$Y''(t) = Y'(t) - 2Y(t)$
$Y'(0) = Y(0) = 1$
with $t_0 = 0$ and $h=0.2$
So what I did:
First ...
0
votes
0answers
42 views
Geodesic equation for a 2D manifold
I am having trouble understanding how the following statement (taken from some old notes) is true:
For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$
If we assume that $$\dot x^a\dot ...
0
votes
1answer
25 views
Differential equations of first order?
I have the equation y' - 2xy =2x*e^(x2) (1)
To solve it,I do y' -2xy=0 then lny=x^2 +c so e^(x^2 +c)=y
I find y' here so I have y'=(2x+c') * e^(x^ 2 +c)
I replace this in equation (1) and I have ...
1
vote
1answer
58 views
How to solve a tensor differential equation?
Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$
where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.
The original Problem
How does ...
2
votes
1answer
79 views
how do I prove this inequality involving ODE solutions
I have the following equations:
$$
y' = (x^4 + y^4)^{1/4} , \quad y(0) = 1.
$$
I need to prove that:
$$e^x\le y(x) \le 2e^x - x.$$
1
vote
3answers
38 views
Solution of system of linearly dependent equations.
So, I have the system of equations $x'(t) = Ax$ where $A$ is first row-(4,-2) and second row - (8,-4). This has two eigenvalues, both are 0. But I tried to solve it this way:
$x_1' = 4x_1 -2x_2$
and
...
0
votes
1answer
38 views
Solving $y'=2y+3$
I am struggling because I am dealing with a differential equation that leaves me a bit confused:
$y'=2y+3$
My attempt to solution
As $y=\frac{dy}{dx}$ then $\frac{dy}{dx}=2y+3$ so
$$\int ...
1
vote
1answer
31 views
Show D.E. is exact
Show that the differential equation $$(y^3 + \cos(t))y'= 2+ y\sin(t)~,~~~~ y(0)=-1$$ is exact and solve the i.v.p.
So far I have $$(y^3 +\cos(t))y'- y\sin(t)-2=0$$ where $M_y=N_t,$ making it exact. ...
2
votes
1answer
45 views
differential equation12
After 30 days of radioactive decay,100 mg of a radioactive substance was observed to remain. After 120 days, only 30 mg of this substance was left.
A. How much substance was originally present?
B. ...
1
vote
2answers
31 views
Proving the Laplace Transform
By using the integral definition of the Laplace Transform I need to prove that:
$$L\left({t}^n{e}^{at}\right) = \frac{n!}{(s-a)^\left(n+1\right)}$$
So far I got:
$$L\left(t^ne^{at}\right) = ...
1
vote
0answers
22 views
Necessity for Osgood's Uniqueness Theorem
Let $\phi (z)$ be continuous and increasing function in the interval $[0,\infty)$, $\phi (0)=0, \phi (z) > 0 \, \forall z > 0$ with also:
$$\lim\limits_{\epsilon \rightarrow ...
1
vote
1answer
38 views
3 tank mixing problem
There are 3 tanks filled to capacity with fresh water, all with a 100 liter capacity. At t=0, brine with .5 kg/l salt concentration flows into tank 1 at a 3 l/min rate. The other flows are:
tank 1 -> ...
1
vote
1answer
67 views
Differential equation with no constants.
Is there a way to calculate $\psi$ which is a function of $x$ out of this differential equation:
$$
\frac{d^2 \psi}{d x^2} = x^2 \psi
$$
0
votes
0answers
24 views
Help solving the (degenerate) SDE: $X_t =\int_0^t |X_s|^\alpha ds$
In a homework exercise I am, as an example of non-uniqueness of SDE's with drift only Hölder continuous of index in (0,1) , asked to show that
both the zero process and $X_t=C\cdot t^p$ where ...
1
vote
1answer
56 views
Finding three independent solutions for the system
I'm stuck on this assignment. Not sure how to begin.
Let $\begin {bmatrix}2&1&1\\ 0&2&3\\ 0&0&2\end {bmatrix} = A$ as part of the system $Ax=x'$. Find three independent ...
1
vote
2answers
89 views
General solution of a differential equation $x''+{a^2}x+b^2x^2=0$
How do you derive the general solution of this equation:$$x''+{a^2}x+b^2x^2=0$$where a and b are constants.
Please help me to derive solution thanks a lot.
2
votes
1answer
33 views
How do I find the solution $y(x)$ for $0 \le x \le \pi$ for this boundary value problem?
$\displaystyle y''+\frac{1}{25}y=\sum \frac{1}{n} \sin(nx)$
$y(0)=y(π)=0$
I know I'm supposed to use $y(x)=\sum B_n \sin(nx)$, and the answer is supposed to be
$\displaystyle ...
3
votes
1answer
118 views
Lemme 2.4 in Morse theory by Milnor
This is lemma 2.4 from "Morse theory" by Milnor ,with the prove
I have some questions about this prove :
1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
5
votes
1answer
41 views
Solving differential equation from Cauchy problem
I am getting acquainted with the Cauchy equations and I am trying to solve an exercise, taking the examples from my class notes. The exercise is:
$$\begin{cases} y'=xy+x\\y(1)=2
\end{cases}$$
I have ...
0
votes
0answers
45 views
Homogeneous ODE
I want to thank you for helping me with my homework. The question is:
Solve the homogeneous ordinary differential equation by matrix. Trial solution is: $λt$
...
0
votes
1answer
33 views
Show that unhomogeneous linear differential equation degree $n$ has $n+1$ roots independent linearity on $(a,b)$
Show that non-homogeneous linear differential equation degree $n$ has $n+1$ roots independent linearity on $(a,b)$ with coefficients in equation are continuos funtions on $(a,b)$.
0
votes
0answers
17 views
Coordinates and Change of Basis, Applications of Vector Spaces
Is the sum of two solutions of a homogeneous linear differential equation also a solution
0
votes
2answers
54 views
Initial value problem uniqueness (Lipschitz)
Show that each of the following initial-value problems has a unique solution ($0 ≤ t ≤ 1 , y(0) = 1$).
$$y' = \exp(t-y)$$
Theorem 1: Suppose that $D=\{(t,y)|a≤t≤b, −∞< y<∞\}$ and that $f(t,y)$ ...
1
vote
0answers
76 views
Relating terms in differential equation with power series
Having problems with a task on a differential equation containing a power series.
Given
$$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$
$$\frac{dy}{dt} = \lambda y$$
$$x(y) = y + ...
1
vote
2answers
32 views
Formula for the pseudofrequency using approximations
We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}(u^2)$
I need to show that $\frac{w_1}{w_0}=1+\frac{\lambda}{N^2}+\mathrm{O}(\frac{1}{N^3})$ such that ...
2
votes
2answers
50 views
How to solve the differential equation of second order?
How to solve the differential equation
$y \dfrac {d^2y} {dx^2} -{(\dfrac {dy} {dx}} ) ^2=0$ .
thank you for your time.
4
votes
3answers
89 views
Second order linear homogeneous ODE with constant coefficients
In homework I was asked to find all solutions to the following ODE:
$$x''+ax'+bx = 0$$
After reading, I know the following.
(a)
If $t^2+at+b = (t-\lambda_1)(t-\lambda_2)$ with ...
3
votes
1answer
37 views
Poicare-bendixon show periodic solutions.
Show that the system $x^{'}=x-y-x^{3}$,$y^{'}=x+y-y^{3}$ has a periodic solution.
went to polar.
r$r^{'}=x^{'}x+y^{'}y$
thus
r$r^{'}=x^{2}-x^{4}+y^{2}-y^{4}$
collecting plus squares.
...
2
votes
1answer
70 views
Some Chaotic Fun for the middle of the night. (basic)
This question involves some simple ideas from the Lorenz equations when $\sigma=1$
Otherwise know as the chaotic waterwheel.
Basically the question is to show that the moment of Inertia in the ...
2
votes
0answers
50 views
Bifurcation in 3 dimensions (simple)
I am Doing a project i have a toy system that describes a bifurcation in 3 dimensions i am posting this in part because i can no longer understand what i have written down ( its been awhile) i have ...
1
vote
1answer
62 views
How to solve simple differential equation
Solve the initial value problem
$$y' = \frac{1 + y^2}{x};\ y(1)=1;\ x>0$$
by separation of variables.
0
votes
1answer
45 views
Find the solution to this 4-by-4 matrix ODE
Find the solution to the system $y'=Ay$, where A is given by:
$A=
\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & -1 & 3 \\
0 & 0 ...
1
vote
2answers
113 views
Picards Method for $\sin 2t$
This is a HW question.
Derive the Taylor series for $\sin 2t$ by applying the Picard method to the
first-order system corresponding to the second-order initial value problem
$x" = −4x; x(0) = 0, ...
2
votes
2answers
49 views
Solution to differential equation
I have a differential equation $$x'=\sin(x).$$ WolframAlpha displays the solution as $$2\cot^{-1}(e^{{c_1}-t}),$$ where $c_1$ should be a constant.
However, I cannot derive it. I would appreciate any ...
1
vote
1answer
45 views
Solution to first order differential equation
How do I argue that the following IVP has no meaningful solution?
$$\frac{dx}{dt}=\sqrt{(x^2-t)},x(1)=0$$
The basic condition for existence of solution is that $\sqrt{(x^2-t)}$ is differentiable and ...
1
vote
0answers
41 views
Planar circular restricted 3-body problem
Hi and sorry for my bad English, it's not my first language. I'm trying to find the equations of motion of the planar circular restricted 3-body problem. I did the gravitational force, but I have some ...
2
votes
1answer
84 views
Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.
Given the function
$$
f(x) = \left\{\begin{array}{cc}
e^{- \frac{1}{x^2}} & x \neq 0
\\
0 & x = 0
\end{array}\right.
$$
show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.
So I have to show ...
0
votes
1answer
59 views
Analytic solution to explicit midpoint rule applied to an ODE
By solving a three-term recurrence relation, calculate analytically the sequence of values {$y_n:n=2,3,4,...$} that is generated by the explicit midpoint rule: $y_{n+2}=y_n+2hf(t_{n+1},y_{n+1})$ when ...
0
votes
1answer
47 views
when does $\frac{dq}{dT} = \frac{Q}{T}$
If I have a current going through a wire, I can say say that:
$$i = \frac{dq}{dT} = \frac{Q}{T}$$
If there is some sort of symmetry or homogeneity. If I were to solve this Differential Equation, I ...
1
vote
0answers
54 views
Sturm-Liouville eigen value problem with one-dimensional eigenspace
Let $p\in C^1([0,1])$ with $p>0$ $\forall x\in[0,1]$ and $q\in C([0,1])$. Define the operator $L: C^2([0,1])\rightarrow C([0,1])$ by
$$
Lu = -(pu')' + qu,
$$
and define $L_{\lambda} = L-\lambda I$. ...
1
vote
2answers
52 views
Exact differential equations. Test to tel if its exact not valid, am I doing something wrong?
I got this differential equation:
$$(y^{3} + \cos t)'y = 2 + y \sin t,\text{ where }y(0) = -1$$
Tried to check for $dM/dY = dH/dY$ but I cant seem to get them alike. So what would the next step be ...
1
vote
1answer
47 views
Boundary-value problem in differential equations
Consider the problem:
$$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$
Find the eigenvalues.
How should one proceed about this problem? I am complete ...
