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

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88 views

Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
2
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1answer
72 views

Theoretical and computational results of ODE differ a lot! Why?

Hello. I have a problem in that my theoretical and practical(computational) calculations differ a lot. in 1 order of magnitude actually. Impulse I needed to jump over the saddle point in one ...
2
votes
1answer
21 views

How can you alter the Volterra-Lotka system to obtain a model of cooperative species?

The Volterra-Lotka system for two competitive species is: \begin{equation*} \frac{dx}{dt} = x(-Ax-By+C) \\ \frac{dy}{dt} = y(-DX-Ey+F) \end{equation*} where $x,y\geq 0 $ and $ A,B,C,D,E,F$ all ...
2
votes
1answer
23 views

Transformed pde but my answer doesn't match solution?

$$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + \frac{du}{dx} + 2\frac{du}{dy} + 3u = 0$$ Let $u = ve^{ax + by}$ and find $a, b$ such that we can transform to the following equation $$\frac{d^2v}{dx^2} + \...
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0answers
22 views

Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
2
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0answers
35 views

ordinary differential equations

I am trying to understand how the solution of this equation goes: $$\frac{y^2-1}{y}\cdot \sin(x^3)=\frac{dy}{dx}$$ with initial condition $y(0)=-0.5$ I would like to understand if the solution can ...
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31 views

Clarification of Fuchs's theorem

Here is Fuchs's theorem My professor has been saying the last couple of classes that if $p(t)$ and $q(t)$ are polynomials, then the second order differential equation converges everywhere. He hasn't ...
2
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1answer
96 views

Extremizing the following boundary value problem

Consider the functional $$J(y)=y^2(1)+\int_0^1y'^2(x)\,dx$$ with $y(0)=1$ , where $y\in C^2[0,1]$. If $y$ extremizes $J$ then find the value of $y(x)$. I tried through Bolza problem. Firstly Euler-...
2
votes
1answer
50 views

Second Order Non-Linear Ordinary Differential Equation

I have the equation $$x_{tt}+cx_t+x=x^2$$ where $c$ is constant and $x=x(t)$. If the $x^2$ wasn't on the right hand side of the equation then I could solve this easily by the method of ...
2
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1answer
141 views

Reduce PDE to ODE

Maybe you don't want to check all the details, but could look at a few equations here. Would you mind leaving a comment that you at least some part looks okay?- This way, I know that at least somebody ...
2
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1answer
392 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
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0answers
167 views

What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, Picard–...
2
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0answers
70 views

General solution of $ty'+2y=4t^2$

Should we left the general solution of the differential equation $t\frac{dy}{dx}+2y=4t^2$ as $t^2y=t^4+c$ instead of $y=t^2+c/(t^2)$ ($c$ is an arbitrary constant)? Does the solution $y=t^2+c/(t^2)$ ...
2
votes
2answers
295 views

How to prove linearity?

Let suppose third-order differential equation, that solved for highest derrivative admits solution: $Y(t) = y(t) + C_1 f_1(t) + C_2 f_2(t) + C_3 f_3(t),$ where $y(t)$ is some solution, $f_1(t), f_2(t),...
2
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1answer
42 views

differential equation power series solution

I am trying to solve this equation using power series $$ (1-x)y"-xy'+y=0 $$ Knowing that $y(0)=-2$ and $y'(0)=6$. Please I need someone's help, I get a relation between $c(n)$,$c(n+1)$, and $c(n+...
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0answers
60 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). $\...
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20 views

Equillibria to Differential Equations

I am wondering what the exact definition is of an equilibrium to a differential equation. It seems like the general consensus implies that a differential equation will only have an equilibrium if it ...
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0answers
19 views

For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
2
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0answers
27 views

Origin/justification of the condition in variation of parameters?

The method of variation of parameters (on e.g. $y"+py'+qy=g$ that yields $y=A(x)y_1 +B(x)y_2$) requires one to use, in addition to the constraint provided by the actual differential equation, one has ...
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23 views

What is a function which differs when differentiated with respect to x and then with y to function differentiated with respect to y and then x? [duplicate]

What is a function which differs when differentiated with respect to $x$ and then with $y$ to function differentiated with respect to $y$ and then $x$? Even if it differs at one particular point.
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1answer
44 views

ODE: $x' = x+x^2+x^3t$, $x(2)=x_0$. Find $\frac{\partial x}{\partial x_0}|_{x_0=0}$

Problem: $x'=x+x^2+x^3t$ and $x(2)=x_0$. Find $\frac{\partial x}{\partial x_0}|_{x_0=0}$ My attempt: Multiplying both sides by $x^{-3}$ and substituting $y=x^{-2}$ gives us Riccati equation: $\...
2
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1answer
50 views

Initial&Boundary Value problem-Fourier

$$u_{t}=u_{xx}, \hspace{5mm} x>0, t>0$$ $$u(0,t)=0 \hspace{3mm} u(x,0)=f(x)$$ We want that the solutions are bounded. We are looking for solutions of the form $$u(x,t)=X(x) \cdot T(t)$$ $X(...
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1answer
57 views

Solving laplace equation and decomposing

I want to solve $$ 9U_{xx}+U_{yy}=\sin (2\pi x) + \sin(2\pi y) \label{eq:1}\tag{1} $$ with $U=0$ on the boundary of the unit square. I know you would have to decompose the problem to satisfy each of ...
2
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0answers
54 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 \hspace{3mm}\textrm{and}\hspace{...
2
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1answer
61 views

Find $dy/dx$ of $(xy^2)+5 = x + 2y^2$

For the solution I got $$\frac{y^2-1}{ 4y-2xy} = dy/dx$$ I just want to know if this is correct. Also it says to evaluate $dy/dx$ at $(1,2)$. Would the solution to that be $3/4$?
2
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0answers
29 views

ODE from systems biology, can I generalize this? Have solution but not sure how to arrive at it.

Reading a systems biology book, and it describes a model with the following ODE: $$ \frac{dY}{dt} = -\gamma Y + v_1 X_1 (T - Y) + v_2 X_2 (T - Y)$$ where $Y$, $T$, $T - Y$, $X_i$, $a$ and $v_i$ are ...
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0answers
64 views

Is there an elegant proof of this elementary bifurcation theory result?

Let's suppose I have a $C^1$ function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $(x,\lambda)\mapsto f(x,\lambda)$. Suppose there is a unique solution of the equation $f(x,\lambda_1)=0$, ...
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0answers
30 views

Find the points at which an IVP admits at least two solutions

Given the IVP: $$\frac{dy}{dx} = x + |\sin(y)|$$ $$y(x_0) = y_0$$ Find the points in $\mathbb R^2$ at which this IVP admits at least two solutions. Clearly, $f(x,y) = x + |\sin(y)|$ is Lipschitz ...
2
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0answers
127 views

Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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0answers
18 views

Find the trajectories that follow drops of water on a given surface.

We have a saddle that have the form of the surface $z=y^2-x^2$ and is outside, under the rain. Find the trajectories that drops of water will follow if they fall on such surface. Can somebody give me ...
2
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0answers
174 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
2
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1answer
38 views

Find a first order DE satisfied by the Wronskian

Consider the homogenous linear DE $y''+p(x)y'+q(x)y=0$. Suppose that $y_1$ and $y_2$ are two solutions on $[a,b]$. Define $W(x)=y_1(x)y_2'(x)-y_1'(x)y_2(x)$. Find a first order DE satisfied by W and ...
2
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0answers
53 views

Differential Equation $(1+x^2)y'-2xy=\cot(x)$ after integrating factor

$$(1+x^2)y'-2xy=\cot(x)$$ or $$y'=\frac{2x}{1+x^2}y+\frac{\cot(x)}{1+x^2}$$ if I use an integrating factor $(e^{\int\frac{-2x}{1+x^2}dx}=\frac{1}{1+x^2})$ I get $$\frac{y}{(1+x^2)}=\int\frac{\cot x}{(...
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0answers
40 views

Separable equation

I am looking at this first order separable differential equation, and I am stuck. Here is the equation: $\frac{du}{dt}=u$ I seperated like this: $ \frac{du}{u}=dt $ Integrated both sides and ...
2
votes
2answers
63 views

Differential equation : $y' = (x+1)/(xy+x)$

So, I have the following differential equation to solve : $$y' = \frac{x+1}{xy+x}$$ After several steps, I get here : $t^2 + 2t = 2x + 2ln(x) + c$ How do I isolate $t ?$ thank you! By the way, $ t=...
2
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1answer
37 views

Setting up a differential equation

The question was: A radioactive element decays at a rate proportional to the mass remaining. Initially the mass is 10mg and after 20 days it is 5mg. Set up a differential equation describing this ...
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0answers
47 views

Solving differential equation with small parameter

I am trying to solve the following equation $$\frac{x}{x+1}\frac{d^{2}\left(\phi^2\right)}{dx^{2}}+\frac{2x+1}{(x+1)^{2}}\frac{d\left(\phi^2\right)}{dx}=\frac{1}{3\phi}$$ with $x\ll1$ and $\frac{d\...
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1answer
62 views

Complex valued trial solution

For this problem, I've gotten the stage $$y'''' + 4y''' + 14y'' + 20y' + 25y = e^{(-1 + 2i)x} = e^{-x}( cos2x +i sin2x )$$ but am I not sure how to proceed from here. Is my trial solution $$e^{-x}(...
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0answers
30 views

Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = \...
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0answers
33 views

Zeroes of a solution to a differential equation

Show that any solution to the equation $y''+xy=0$ has at least 15 zeroes on the interval $[-25,25]$. Please give me a hint.
2
votes
1answer
155 views

Solving second order ODE with polynomial coefficients

What is the solution Can some one solve this..... $T.y''+T'y'+T''y=0$., where $T$ is a polynomial. Like $1-2x-x^2$ etc. in $x$........ Is there anyway to assume the answer as polynomial and go ...
2
votes
2answers
67 views

Using the Laplace Transform solve $y''+6y'+5y=e^t$

The initial conditions are $y(0)=0$ and $y'(0)=1$. I began the process and ended up with $Y=1/(s-1)(s^2+6s+4)$. Since the second factor in the denominator does not factor so I have a feeling I messed ...
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0answers
24 views

Is the following complex value differential equation always has a solution?

Let $a(z)$ be a fixed complex value complex variable function, not necessarily holomorphic. Consider the following differential equation $$ \frac{\overline{\partial}f}{\partial \overline{z}}+af=0. $$ ...
2
votes
1answer
270 views

Leibniz rule for improper integral

We know that the Leibniz integral formula $$\frac{d}{dt}\int_{\phi(t)}^{\psi(t)} f(t,s) ds = \int_{\phi(t)}^{\psi(t)} \frac{d}{dt}f(t,s) ds+f(t,\psi(t))\frac{d}{dt}\psi(t) -f(t,\phi(t))\frac{d}{dt}\...
2
votes
1answer
81 views

Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
2
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0answers
39 views

Differential equation of the form to find

Lets $f(z)$ is some analytic function on complex plane and $y(z)$ is known analytic function on complex plane. Problem statement: find all $f(z)$ that: $$f(z) = f(z\frac{\partial}{\partial z})y(z)$$ ...
2
votes
0answers
57 views

Proving the Bessel function solves the Bessel equation

Using the notation for the Bessel function as $J_n(z)=\sum \limits_{k=0}^{\infty}\frac{(-1)^kz^{n+2k}}{k!(n+k)!2^{n+2k}}$, I want to show that $w=J_n(z)$ satisfies $w''+\frac{1}{z}w'+\left(1-\frac{n^2}...
2
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0answers
167 views

Periodic Solution of Damped Pendulum with Constant Torque

I have a system of ordinary differential equations $ \theta' = v$ $ v' = -bv - \sin \theta + k$ These are the equations for a pendulum with $\theta$ being angular position, and $v$ being angular ...
2
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0answers
179 views

Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
2
votes
0answers
86 views

Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil $$A=-\...