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

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Finding the differential equation

I'm having trouble with question c) as it is asking to find the differential equation for the diameter given the equation $D=aM^{1/3}$ My attempt Why is my differential equation wrong?
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General solution to 3D linear 2nd order PDE using Wronskian and Integrals?

Using the Wronskian and Indefinite Integrals, I can write the solution to the general one dimensional second order linear non-homogeneous differential equation $$ y''+p(x)y'+q(x)y=g(x) \\ y^*(x)= ...
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39 views

Coefficients of spherical solution to Laplace's equation with difficult Robin boundary conditions

I'm trying to solve Laplace's equation in an (axisymmetric) external spherical domain. The controlling equation is: $$\nabla f = 0$$ $f$ must dissappear at infinity, and at the surface of the ...
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28 views

Verify solution to $\frac{i R}{L}+i'=\frac{U_m \sin (\text{$\omega $t})}{L}$

Show that this is a solution $$i(t)\text{:=}\frac{U_m \sin (t \omega -\varphi )}{Z}$$ to $$i'+\frac{i R}{L}=\frac{U_m \sin (t \varphi )}{L}$$ given: $$\varphi =\tan ...
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37 views

How to solve differential equation $f(y^2)\sqrt{y^2+y'^2}-\frac{f(y^2)y'^2}{\sqrt{y^2+y'^2}}=C$

I have the following differential equation I need to solve: $$f(y^2)\sqrt{y^2+y'^2}-\displaystyle\frac{f(y^2)y'^2}{\sqrt{y^2+y'^2}}=C,$$ where $y=y(x), \;y'=y'(x)=\displaystyle\frac{dy}{dx}, ...
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book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
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Solving Laplace equation in polar coordinates

I have some assignments to do and I don't even know where to start. The notes in the course aren't too good, so I didn't understand too much from them. Given $$ \Omega = \{(x, y) \in \mathbb{R}^2 , ...
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Nature of the infinite differential sum operator?

Consider the operator $$ Hf = f + f' + f'' +\cdots = \sum_{i=0}^\infty \left[ \frac{d^i f}{dx^i}\right] $$ I am trying to determine what $ Hf $ is entirely in terms of $f$. I note the following ...
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38 views

Is the calculation of Green's function correct?

I am not sure if all the calculations are correct.Could you check for me please ? ...
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27 views

Variable change for homogeneous equations

I got my brain for some days on this two differential equations with no luck at all, have tried different variable changes but it seem that I always get to a dead end, will be glad if someone could ...
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Invariants under Hamiltonian mechanics?

I am interested in certain properties of measures evolving according to Hamiltonian mechanics. Say we have a point $z$ in phase space: $z = (p,q)$ where $p$ is a generalized momentum vector and $q$ is ...
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D'Alembert operator Green function in arbitrary dimension

I am interested to learn about the Green function for the D'Alembert operator in arbitrary dimensions. While searching through the web I came across the following document: ...
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Evaluating first order differential equation

I was given $$(x+3y^2)\frac{dy}{dx}=y$$ and also $y>0$ so I wrote it as $$\frac{dx}{dy}=\frac{x}{y} + 3y$$ now substituting $x=y.t$ and $$ \frac{dx}{dy}=t+y.\frac{dt}{dy}$$ I am finally left with ...
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Find what values of 'b' have bounded solution(differential equation)?

$y′′ + b^2{y} = f(t)$ $ f(t) = t$ for $0 < t < 2\pi$ ($2\pi$ periodic sawtooth wave) This is my solution to the differential equation. $y(t) = C_1\cos(bt) + C_2\sin(bt) + b^{-3}\left(bt - ...
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Schroedinger equation in power law potential

Consider the one dimensional Schroedinger equation: \begin{equation} (1)\left(-\frac{\hbar^2}{2 m} \frac{d^2}{d x^2} + \frac{1}{2} m \omega^2 |x|^\mu \right) \Psi(x) = E \Psi(x) \end{equation} By ...
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Can a “pseudo” SDE be solved as an ODE

I am not that much into ordinary differential equations. Assume I have a stochastic process satisfying the "SDE" $dX_t = (rX_t + Y_t) dt$ with $Y_t$ preferably just some unspecified SDE. Can it ...
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Differential equation general solution based on discriminant

I need to solve the differential equation $y'' + y' = -2xe^{-x}$. This gives $\lambda^2 + \lambda = 0$, $\Delta > 0$ so the general solution is $$y = c_1e^{\lambda_1x} + c_2e^{\lambda_2x}$$ But ...
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diffrential equation of degrre 3

I need help to solve it. Thanks $$x^3\sin(x)y^{'''}-(3x^2\sin x+x^3\cos x)y^{''}+(6x\sin x+2x^2\cos x)y^{'}-(2x\cos x+6\sin x)y = 0$$
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42 views

Prove theorem of Existence and Uniqueness of solutions of a $n$-th order differential equation using Picard-Lindelöf theorem

In my textbook, after stating and proving the Picard-Lindelöf theorem, the following corollary is stated: Corollary. Let $f:[t_0,t_1]\times\mathbb{R}^n\longrightarrow\mathbb{R}^n$ be a continuous ...
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Problem regarding particular integral based on other ODEs

If $\phi_1(x)$ is a particular integral of $Ly=\dfrac{d^2y}{dx^2}-a\dfrac{dy}{dx}+by=e^{ax}+f(x)$ and $\phi_2(x)$ is a particular integral of $Ly=x^{ax}-f(x)$;a,b being constants, then a particular ...
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Solve the following second order ODE: $4t^2 x x''=3t(3x^2+2)+2(3x^2+2)^3$

How can I solve this second order ODE? $$4t^2 x x''=3t(3x^2+2)+2(3x^2+2)^3$$
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Properties of solutions of system of integral equation.

Assume $g:[0,\infty) \to \mathbb R$ to be continuous and $$\int_{0}^{\infty} s|g(s)| \,\mathbb ds< \infty .$$ I want to find $\alpha>0$ such that the system of integral equations ...
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Proving that an equilibrium point is globally asymptotically stable

I have a nonlinear, autonomous, dynamical system (system of eight ODEs). I have shown that the equilibrium point is locally asymptotically stable. I want to show that it is globally asymptotically ...
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Conditions to guarantee unique limits of trajectories.

For a real function $f$ on $\mathbb R^n$, such that no trajectories of the gradient escape to infinity, what are necessary and/or sufficient conditions so that each trajectory limits to a unique point ...
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Show $k$-form/chain identity

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k\rightarrow\mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show that ...
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Uniqueness of backward heat equation on closed manifold with given initial data

Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation $$\frac{\partial f ...
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Could anyone help check the derivation?

I'm reading this paper and I'm trying to derive the gradient equation in (5) of the paper. But I couldn't get the right answer. My derivation is as follows. Could you help check it please?
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Finding the index of a linear vector field at the origin

For the linear vector field $f(x,y) = (f_1, f_2) = (ax+by,cx+dy)$, show that the index with respect to the origin is $\pm 1$ depending on whether $ad-bc > 0$ or $ad-bc < 0$. I've gone ...
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Partial Differential Equations Course And Differential Geometry Prerequisites

Is the ordinary differential equations course a prerequisite for the partial differential equations course for a person who has passed the integral calculus course? Is it really required to have ...
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How to solve $4x^2\cos y\sin y\partial{y}-3x\sin y\partial{x}+8\sin^2y\partial{y}=0$?

$$4x^2\cos y\sin ydy-3x\sin ydx+8\sin^2ydy=0$$ find the solution of this Bernoulli equation. I dont know where to start
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a question regarding wronskian

I was working on following problem: Let $y_1$ and $y_2$ be solutions of $x^2y'' + y' + (\sin x)y = 0$ satisfying $y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $. I worked like following: since ...
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Limit to infinity from a differential equation

Let $R'(t) + \nu R(t) = \nu F(t)$, $F(0)=0$, $R(0)=0$, $f(t) \geq 0$, $F(t) = \int_0^t f(\tau)d\tau$, $F(t) \leq 1$, and $\lim_{t \rightarrow \infty} F(t) = 1$. I solved the differential equation and ...
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Find the general solution for the following non-homogeneous equation?

$y^{(4)} - 2y''' + y'' = 2e^{3x} + x$ Attempt: The characteristic equation is $r^4 - 2r^3 + r^2 = 0$. ==> $r^2 (r^2 - 2r + 1) = 0$ ==> $r^2 (r - 1)^2 = 0 $ ==> $r = 0, 0, 1, 1$. So, $$y_h = ...
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Find a Liapunov function to show asymptotically stable

Consider the system: \begin{cases} \dfrac{dx}{dt} = y \\[12pt] \dfrac{dy}{dt} = -(1+x^{2})\,y-\sin(x) \end{cases} $(0,0)$ is a critical point of this system and I need to show that it is ...
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how to solve Schrödinger equation

I would like to solve a complete solutions of the Schrödinger equation for a particle with time & position dependent mass ($m(x,t)$) moving in a potential $V(x,t)$. Any suggestions to solve ...
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Reducing a system of differential equations

Let $\mathbf F$ be a system of 1st order differential equations in $n>3$ variables $$\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^n$$ $$\frac{d\mathbf{u}}{dt} = \mathbf{F}(\mathbf{u})$$ such that ...
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method of characteristics in a nutshell

Good morning everybody, I need a quick reference for the following inhomogeneous first-order pde... namely $$f(x,y,z)=A\partial_x\varphi+B\partial_y\varphi+C\partial_z\varphi,$$ where $\varphi\in ...
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Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): Here is $\alpha, \beta, \gamma, \delta$ - some constants, $u$ - control variable. ...
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show that the bifurcation occur at an exact point

Hi I'm having trouble with this problem. I have everything completed, but not the parte where I have to show that the saddle node bifurcation occur at k= -a + 1/2(a)^(1/2) and k= -a - 1/2(a)^(1/2). I ...
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Particular solution of system of differential equations

Solve system of differential equations $$\begin{cases} x'(t)=2y(t)-x(t)+1 \\ y'(t)=3y(t)-2x(t). \end{cases} $$ My solution: First, I find the characteristic values $$\begin{vmatrix} 3-\lambda & ...
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About Wronkian and Able' s identity

I want to what is the condition on $p_1$?? And I want to ask is the wronkian of any 2 fundamental solutions totally depends on $p_1$?? Because any fundamental solutions can be different from a ...
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Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
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given a solution of a second order ODE, what is the way to find another linearly independent solution?

So if i'm given a second order linear diff eq. and one of its solutions, what is the way to find another linearly independent solution? Thanks in advance!
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Second order differential equation with unknown coefficient numerically

I have the following differential equation $$x''(t)+p(t) x'(t)=0,\qquad t\in[0,1]$$ I need to solve it numerically (find $x(t_i)$, where $\displaystyle t_i=\frac{i}{N}$ for $i=1,...,N-1$) using next ...
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Linear second order PDE with constant coefficients

I am doing this mathematical problem $c*G_{vv}+ d*G_{u} + e*G_{v} +f*G=0$, where $c, d, e$ and $f$- are constant coefficients. I already know that this is second order PDE and we classify it by ...
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Using Fourier Analysis to determine Green's Function of Laplace's equation

I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have ...
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Solving ODE by finite differences and Newton's method.

Given this boundary value problem $y'' = (x^2(y')^2 - 9y^2 + 4x^6)/x^5, \quad 1 \leq x \leq 2, \qquad (1)\\ y(1) = 0, \; y(2) = \ln 256$ I have to solve the problem using finite differences, for 21 ...
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Ode with Piecewise function

We can write this $$12x"+36x'+48x=f(t)$$ my main problem is how to solve this non-homogeneous ODE I know how to do this as 2 different ode unfortunately its not in a syllabus which doesn't use ...
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Can you give some information for rothe method

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
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A differential equation of second order with polynomial coefficients

I have a second order differential equation with polynomial coefficient as follows: $$x^2(x^2-1)y''-x(x^2+1)y'+(x^2+1)y = 0$$ If I have one answer I can clearly obtain the other one. Can you please ...