Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

learn more… | top users | synonyms (1)

0
votes
0answers
20 views

vector space differential equations

Hi! I am working on some differential equations homework and we are up to the linear algebra part. This particular homework set on Vector space is due, but my teacher has not taught the material yet ...
0
votes
0answers
29 views

Showing a system is fully self adjoint for general unmixed boundary conditions

I have been asked to look at the following questions and I'm struggling to solve it. Let $Ly=a_2(x)y''(x)+a_1(x)y'(x)+a_0(x)y(x) , a<x<b$ such that $L^*=L$. i.e. $L$ is a self adjoint linear ...
1
vote
0answers
8 views

Simplifing a Cauchy product to find the recurrence relation when solving a differential equation using a power series solution.

I'm having trouble finding the recurrence relation of the following non linear differential equation: $y''(x)+p(x)y'(x)+y^2(x)=0$ with $y(0)=1$ and $y'(0)=0$ where ...
0
votes
0answers
11 views

solving three first order differential equations simultaneously with varying coefficient

I need to solve 3 first order differential equations simultaneously. I can solve this equation when [A] is constant. But in this case, as I will explain, [A] is function of z. By omitting the uz, I ...
0
votes
0answers
33 views

What is the equation family of the projectile-motion-with-air-resistance eqn?

The general form of the equations of projectile motion with air resistance are (from here) $s_y(t) = -\frac{mg}{k}t + \frac{m}{k}(v_{yo} + \frac{mg}{k})(1 - e^{-\frac{k}{m}t})$ and $s_x(t) = ...
0
votes
0answers
34 views

Invariants of a nonlinear ODE

Given a nonlinear ODE and a simple constraint $x \leq c$ for some constant $c$, how can we describe the largest set (or an approximation thereof) such that if the initial value of the solution of the ...
0
votes
0answers
12 views

How could one go about constructing this relatively simple contagious diffusion-reaction model?

How could one go about constructing a contagious diffusion-reaction model showing the relationship between disease (e.g. Ebola) and number of available healthcare workers in an unevenly distributed ...
1
vote
0answers
14 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
1
vote
2answers
23 views

What is the correct answer to this diffferential equation?

[Question] When solving the differential equation: $$\frac{\mathrm dy}{\mathrm dx} = \sqrt{(y+1)}$$ I've found two ways to express $y(x)$: implicitly: $2\sqrt{(y + 1)} = x + C$ or directly: $y = ...
0
votes
0answers
26 views

Showing a second order DE has characteristic equation

Verify that $y''-2py'+p^2y=0$ has characteristic equation $(m-p)^2=0$ and has solution $y=e^{px}$ I began by trying to solve $r^2-2p+p^2=0$ but I'm kind of stuck where to go. Any help would be ...
0
votes
0answers
25 views

I still could not figure out

IT is our homework problem but I have already submit it. Today, I asked professor, but I still could follow what he said clearly. $\frac{dX}{dt} = \mu(x)$ and $X(0;x) = x$, where $x,X\in R^n$ For ...
0
votes
0answers
30 views

How do I solve this calculs problem [closed]

a) Find the general solution of $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 0.$$ b) Solve $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 8\cos 2t + 6\sin 2t.$$ with $y(0) = 4$, $y'(0) = 0 $ How ...
1
vote
3answers
32 views

The limit of a solution of the logistic equation as time tends to infinity

$$ \frac{dP}{dt} = 3P(4 - P),\quad P(0) = 2.$$ What value does $P$ approach as $t$ gets large, ie. as $t \to\infty$. How do I solve this? Is the idea to this question to first rearrange the equation ...
1
vote
0answers
42 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
0
votes
0answers
21 views

Differential equation Worded Problem [duplicate]

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
1
vote
0answers
19 views

Second Order Differential Equations - Undetermined Coefficients

When solving for this one: $y''-3y'-4y=e^{-x}$ For the trial function, let: $y=Ae^{-x}$ $y'=-Ae^{-x}$ $y''=Ae^{-x}$ $=> Ae^{-x}-3(-Ae^{-x})-4(Ae^{-x})=e^{-x}$ $=> ...
0
votes
1answer
18 views

Differential equation with modulus

I have a problem with $$y-xy'=(ln|x|+1)y^2 $$ because I do not know how to deal with the absolute value. I divide $\frac{y - xy'}{y^2}=ln|x| +1$, then substitute $t' = (\frac{x}{y})'$ get ...
0
votes
0answers
30 views

Weight function in orthogonal polynomials

My problem is how we can obtain the weight function in orthogonal polynomials? As we all know, orthogonal polynomials are defined through below equation. That is the integral:$$\int_a^bf(x)g(x)w(x)\ ...
2
votes
1answer
26 views

Classification of pde

I got stuck on the following problem: Determine the subsets of $\mathbb{R}^2$ where the pde $$u_{xx}+2xu_xu_{xy}+yu_{yy}+yu_x=1$$ is elliptic, hyperbolic and parabolic respectively. Now, at first I ...
0
votes
1answer
47 views

How does this integration make sense?

I simply don't understand how integration can lead from: $ds^2 = a^2(t) \frac{dr^2}{1 - kr^2}$ to $s(r) = \frac{\sin^{-1}(\sqrt{k}r)}{\sqrt{k}}$ I appologize, I've never been quite capable of ...
0
votes
0answers
15 views

Existence and uniqueness of SDEs depending on the expected value?

I was thinking of general mean-field SDEs. But let us just look at something really simple: $$dX_t = dt + dB_t, \quad X_0=x$$ the solution to this SDE exists in a strong sense and is: $X_t = x + t ...
2
votes
0answers
41 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
2
votes
2answers
401 views

Can anyone explain why this equation using the fundamental theorem of calculus works?

\begin{align} \left| f(b)-f(a)\right|&=\left| \int_a^b \frac{df}{dx} dx\right|\\ \ \\ &\leq\left| \int_a^b \left|\frac{df}{dx}\right|\ dx\right|. \end{align} I do not ...
1
vote
0answers
40 views

List of eigenvalues for the Schrödinger equation

I'm writing an algorithm which computes the eigenvalues $E$ of the Schrödinger equation with potential $V(x) = x^2$, ie the harmonic oscillator. The equation is defined as follows $$ y''(x) = ...
2
votes
0answers
26 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
0
votes
0answers
25 views

Conjugacy of linear systems with one zero eigenvalue

I have a question from Hirsch, Smale, and Devaney's "Differential Equations, Dynamical Systems and an Introduction to Chaos." Consider all linear systems with exactly one eigenvalue equal to 0. ...
0
votes
1answer
28 views

integrating a differential equation with two derivatives

How can I solve for y(t) in terms of x(t)? Consider the following diff equation 2y'(t) + y(t) = 2x'(t) - x(t) Thanks edit: I need the solution in terms of ...
0
votes
0answers
11 views

To get a particular solution of Poisson's equation

How can I get a particular solution of this Poisson's equation: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\phi}{\partial r})+\frac{\partial^2\phi}{\partial ...
3
votes
2answers
52 views

Finding function $f(x)$

How do we find the function(s) $f(x)$ given that $$f(x)=\int_{0}^{x} te^tf(x-t) \ \mathrm{d}t$$ My Try : I first used the property $\int_{0}^{a}g(x) \ \mathrm{d}x=\int_{0}^{a}g(a-x) \ \mathrm{d}x$ ...
0
votes
1answer
26 views

Frobenius series method

Can someone use Frobenius series method to solve this differential equation step-by-step for educational purposes? $$x^2y''+(x^2+\dfrac{5}{36})y = 0$$ Thanks in advance.
0
votes
1answer
36 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
0
votes
2answers
27 views

Finding a particular solution of an inhomogeneous ODE

How can one find a particular solution of $$y'' = 120x^4 + 180 x?$$ I assumed $$Y_p= Ax^4+ Bx^3 + Cx^2 + Dx + E.$$ I am not able to find $D$ and $E$.
0
votes
2answers
28 views

First Order Linear Differential Equations: Solve dy/dx = x+ 2y

$\frac{dy}{dx} = x + 2y$ My attempt using the method described in the textbook "Thomas Calculus": $$\frac{dy}{dx} - 2y = x$$ $$P(x)= -2$$ $$Q(x)= x$$ Integral of $-2\, dx = -2x$. Then take the ...
0
votes
1answer
28 views

Help with a differential equation?

I'm confused about what the question is asking. I solved the following equations: $$y'' + 4y = 0 \implies y = c_1\cos2x + c_2\sin2x$$ $$y'' + 4y = \sin x \implies y= c_3\cos2x + c_4\sin2x + ...
1
vote
2answers
35 views

differentiate $y=\sin(xy)$

so I am using chain rule to differentiate this and get down to $ \cos(xy) \times \left( x \frac{dy}{dx} + y \right)$ and then I don't know what to do next. The book says the answer is $\frac{ ...
0
votes
2answers
22 views

Optimization of a rectangular container

A rectangular sheet of tinplate is $2k$ cm by $k$ cm. Four squares, each with sides $x$ cm, are cut from its corners. The remainder is bent into the shape of an open rectangular container. Find the ...
1
vote
1answer
27 views

Solve ODE by substitution

Solve the differential equation by using an appropriate substitution. $(x-y)dx+xdy=0$ So let $y=ux$. Then $dy=xdu+udx$. Plugging in we get $(x-ux)dx+x(xdu+udx)=(x-ux)dx+x^2du+uxdx=xdx+x^2du$ I've ...
1
vote
0answers
15 views

Solving a system of 3 DE's. Need a tip for finding eigenvecotr

Hello I have this system: $$ x'=2kx + ky + kz, y'=kx+ 2ky + kz, z'=kx+ ky+ 2kz $$. I found that λ=k and 4k. I am solving for the first eigenvector when λ=k, and end up with this: a+b+c=0, can anyone ...
1
vote
1answer
25 views

Solve the differential equation for y(t) in terms of x(t)

Consider the following differential equation: $$ (t+1)y'(t) + y(t) = x'(t) + x(t) \quad (t > 0) $$ With initial conditions $$ y(0) = 0 \quad x(0) = 0 $$ Solve for $y(t)$ in terms of $x(t)$. ...
0
votes
0answers
27 views

Lyapunov-Schmidt reduction.

Use Lyapunov-Schmidt reduction to find an expression, or approximation, of the set of equilibria, as a function of the parameter $\lambda$, of the planar vector field ...
0
votes
2answers
29 views

How to solve second order linear ODE with variable coefficients?

How to solve $t^4\frac{d^2x}{dt^2}+2t^3\frac{dx}{dt}+x=0$ with $x(\pi/2)=0$? I know that the solution is $x(t)=c_1\sin(1/t)$ but I don't know the way to find it.
0
votes
2answers
75 views

What is the general solution for $y''e^{-y} =1$? [closed]

how can I find the general solution for an ODE $$y''e^{-y} =1?$$ Thanks.
1
vote
3answers
52 views

2. Differential equation with initial condition

Does my work look correct? $\frac{dy}{dt}=-3(y-1)$ with $y(2)=-3$ $$\frac{dy}{dt}=-3(y-1)$$ $$\frac{1}{y-1}dy=-3dt$$ $$\int\frac{1}{y-1}dy=\int-3dt$$ $$\ln|y-1|=-3t+C$$ $$\ln|y-1|=C-3t$$ ...
1
vote
1answer
27 views

Solving the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{y+2xy^2}{3x^2y^2+x}$

How do we find the solutions to the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{y+2xy^3}{3x^2y^2+x}$$ It is not homogeneous, neither is it a Bernoulli differential equation. I have ...
0
votes
1answer
19 views

Solving Differential equation $y"+2y'-3y= x-4e^{-3x}$

solve the equation $$y"+2y'-3y= x-4e^{-3x}$$ In this i have solved for particular solution but since $3$ is a root of homogeneous equation $y"+2y'-3y=0$ i am not able to find the value of coefficient ...
0
votes
3answers
52 views

Solving the differential equation: $f(x)yy'=(y')^2-0.5$

I am trying to solve this equation: $f(x)yy'=(y')^2-0.5$ I have already tried traditional methods... Any ideas?
1
vote
3answers
83 views

Fixed points of: $\dot{x}=\sin(y) \qquad \dot{y}=\cos(x)$

How can you find the fixed points of this system: $\dot{x}=\sin(y)\\ \dot{y}=\cos(x)$ Normally I would suggest that you find the points when both functions are equal to 0.
0
votes
2answers
35 views

Two different results of Fourier Transform $xe^{-x}$

I have a function $f$ defined by $$f(x)=\begin{cases} xe^{-x} \textrm{ if } x>0,\\ 0,\textrm{otherwise}. \end{cases}$$ I wish to know the Fourier transform of $f$, i.e, $${\cal ...
0
votes
0answers
33 views

Mathematics Model for measuring the evenness of a distribution

At time $t$, the distribution for a dynamical model is: $a_1(t), a_2 (t), a_3 (t),…, a_n(t)$ as the system evolves it may be expected that if the number of samples in a species is less than the ...
0
votes
1answer
22 views

differential equation using series expansion

Trying to solve xy'= xy + y using the series y(x) = $\sum\limits_{i=0}^\infty a_nx^n$ This is what i have so far. y'(x)= $\sum\limits_{i=0}^\infty na_nx^{n-1}$ xy' - xy - y = 0 x ...