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

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2
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0answers
65 views

Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
1
vote
0answers
16 views

Graphing a Phase Plane

I understand how to attain the fact that the stable and unstable manifold are just linear multiples of the corresponding eigenvectors, however I am confused as to what I am actually plotting in this ...
0
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1answer
56 views

Differentiation of multivariable function proof

I'm looking for the differentiation of multivariable function integral $$\frac{\mathrm{d} }{\mathrm{d} x} \int_{v(x)}^{u(x)}f(t,x)dt=u'(x)f(u(x),x)-v'(x)f(v(x),x)+\int_{v(x)}^{u(x)}\frac{\partial ...
1
vote
1answer
22 views

Solving $x' = Ax$ for real $x$ where $A$ is a matrix with complex eigen values

I have the following linear differential equation system: $$x' = A x$$ where $$ A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & -2 \\ 2 & 2 & 1 \end{array} \right) $$ I ...
0
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1answer
41 views

characteristic equation of differential equation

Given $x''+3x'+2x=4.$ ($''=2nd $ derivative, $'=1st$ derivative) Determine the characteristic equation of this differential equation. I'm having a hard time doing this because of that $4$. Any help ...
1
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0answers
18 views

Simple differential equation modelling question.

The question is: A chemical dissolves in water at a rate equal to 10% of the amount of undissolved chemical per hour. At time $t$ hours the amount of undissolved chemicalis $x$ grams. Initially the ...
1
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0answers
18 views

System of ODEs and DAE system

Let us consider the following system of ODEs: $$ y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0 $$ and the following one: $$ y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0. $$ $f$ and $g$ ...
3
votes
1answer
31 views

Method of Variation of Parameters - Assigning zero works?

I have yet to find a decent answer on this, and so I don't think this question is inappropriate. Also, this question is mainly meant for people that are very familiar with this method. In the method ...
0
votes
1answer
19 views

Stability of system of nonlinear differential equations

In order to find the stability of a nonlinear system of differential equations (in the real plane) we need to show that the eigenvalues of the linearized system are all negative. Can someone explain ...
0
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0answers
47 views

Blowing-up a singular point

I have this system of ODEs: $$x'=-y+ \mu x(x^2+y^2)$$ $$y'=x+ \mu y(x^2+y^2)$$ I already find that in $\mathbb{R}^2$ the only singular point is $(0,0)$. So I have to blow-up the singularity to find ...
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0answers
17 views

A solid rocket model: a differential equations set with ending time unknown

I am modelling a rocket model. Consider a solid rocket motor, (let us for sake of simplicity assume that the propellant is distributed in the case with a cylindrical shape: see shape in fig.1 of the ...
0
votes
0answers
19 views

How to find initial value problem exist?

How can I find the initial value problem $(x^2 - 1) dy/dx = 4xy$ exist in the range of $R={(x,y)||x-1|<1,|y|<1}$ , $y(1)=0$?
0
votes
1answer
29 views

Fastest way to compute minimal polynomial (for solving $x' = A x$, $A$ matrix)

In general, given a $3\times 3$ or $4\times 4$ matrix $A$ which doesn't have a lot of $0$ entries, what is the fastest or less error prone way to compute its minimal polynomial? More generally, I ...
0
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2answers
33 views

Solving $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$.

Given that $u,v$ are functions of $t$, $R$ constant, solve $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$. When trying to find geodesic on cylinder, I get this ...
1
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0answers
32 views

How to find this common solution?

I have the system $$ \ddot{x}=-\frac{k_1 x}{m},~~\ddot{y}=-\frac{k_2 y}{m} $$ with $k_1,k_2,m > 0$. How can I solve this? Can you give me a hint? Edit My idea is to write it this way: $$ ...
0
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2answers
32 views

A small but tricky derivation

I have the dimensionless differential equation $$ \frac{\text{d}^2\phi(s)}{\text{d}s^2} = \sinh(\phi(s)). $$ Apparently, I should be able to derive that $$ \frac{\text{d}\phi(s)}{\text{d}s} = \pm 2 ...
2
votes
1answer
56 views

Diferential equation solution satisfying $y(0) = \pi$

I have the following question from a past exam: Show that the differential equation $\frac{dy}{dx} = \cfrac{e^x + x}{\sin y + 2}$ has a solution satisfying $y(0) = \pi$ What I have done: $$\int ...
0
votes
0answers
31 views

Solving solely continuous system of ode's with matlab

I'm working with the numerical integration of the system of differential equations, $\dot{x}=f(x)$ with the vectorfield, $f(x)$ being solely continuous. Examples of the systems which I'm working on ...
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0answers
15 views

Local Picard Theorem

In order to apply the local Picard theorem in this question, one of the conditions I must satisfy is that $|f|\leq M$. Where can this be inferred from?
2
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1answer
56 views

Quick question about solutions of $y'-y+y^8=0$

I have one small question about solving $y'-y+y^8=0$ but I don't know where. Here's what I do: Let $u(x) = y(x)^{1-8} = y^{-7}$, then $y=u^{- \frac{1}{7}}$ $\frac{du}{dx} = -7 y^{-8} \frac{dy}{dx} = ...
5
votes
0answers
85 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
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1answer
32 views

Locally Vs Globally Lipschitz Confusion

Is there any difference in a function being locally Lipschitz on $\mathbb{R^n}$ and being globally Lipschitz?
3
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3answers
100 views

Need help with simple system of differential equations

thanks to your help I advanced in computing differential equations, but now I encountered another problem I need help with - this time it is a system of differential equations: $$x_1'=-x_2$$ ...
2
votes
1answer
63 views

Maximal Solution of an IVP

I am able to do Q1 it's just the result is required in Q2. I'm having issues framing the proof of Q2 and am struggling with its conclusion . My attempt: Let $J(x_0)$ be the union of all open sets ...
6
votes
1answer
109 views

How to solve the non-linear differential equation $y''=x-y^2$?

$y''(x)=x-y^2(x)$ I'm particularly interested in solutions when $x>0$. I've performed asymptotic analysis and reached the conclusion that solutions must behave as $\pm\sqrt{x}$ when $x\rightarrow ...
1
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1answer
29 views

Applying limit inferior

In the solution to this question while attempting to prove by contradiction it states that if we assume $|x(t)|\nrightarrow \infty $ then $M:=$lim inf $_{t\uparrow b}|x(t)|$ Why can I just not use ...
0
votes
1answer
19 views

Non autonomous lipschitz condition

I can show that the two solutions satisfy the IVP. To show that this does not contradict the existence and uniqueness theorem I am trying to show that the function is not lipschitz. So: ...
0
votes
1answer
23 views

Stability of solution ODE with parameters

For which $a,b \in \mathbb{R}$ the solution $x=y=0$ is stability for system of ODE: $x'=ax+y+x^2$ $y'-x^2+by+y^2$ In some cases it's easy, because we can ...
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0answers
15 views

Numerical solution of ODE

I have a general question about numerical solution of ODE. I want to solve a ODE on an interval where two solutions can exist and intersect. As far as I understand a numerical solution will give the ...
0
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1answer
54 views

Dynamical systems, conjugacy

Consider a family of dynamical systems generated by equations: $y'=ax+b, \ \ a,b \in \mathbb{R}$. Is it true that in this family: 1) There are 4 types of phase portraits up to topological ...
3
votes
1answer
31 views

Questions about solutions of $x'=f(x^2)$

Let $f:\mathbb{R} \rightarrow \mathbb{R}$, class $C^1$ and consider $x'=f(x^2)$ Can all solutions be strictly decreasing? Is every solution either constant or strictly monotonic? I don't know how ...
0
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1answer
25 views

Classify stationary points

I came up with two stationary points in a system of three differential equations. To classify them, I calculated the eigenvalues of the Jacobian, evaluated in this stationary points. For the first ...
2
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2answers
38 views

Singular Points in first order differential

In a first order differential where $y'=f(x,y)/g(x,y)$ why is any point that have $g(x,y)=0$ is not considered a singular point? I mean isn't singular point the points at which $y'$ is not defined?
0
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2answers
29 views

Unique solution: $f(x)$ not globally lipschitz

I don't understand the part of the question underlined in green. I would normally answer this by showing that $f(x)=1+x^2$ is globally lipschitz but it is not.
1
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1answer
33 views

Solutions tending to infinity in finite time both forwards and backwards

Is my solution correct for the part underlined in green? $$x(t)=\frac{\tan(t)+x_0}{1-x_0\tan (t)}$$ so solutions go to infinity at: $$\tan(t)=\frac{1}{x_0}$$ Where $t_2<0<t_1$
-1
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0answers
18 views

Lipschitz property to show existence

I don't have the solution to the part underlined in green. I understand how to show $a<x(t)<b$
0
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1answer
18 views

Solve ODE of plane pendulum

I have seen in a book this system of ode's: $$ \begin{cases} \dot x =y\\ \dot y=-\sin x \end{cases} $$ and the say that the solution is $$ \begin{cases} x(t)=\pm 2\arctan(\sinh(t)) \\y(t)=\pm ...
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0answers
20 views

$\gamma(t)$ is not asymptotically stable unless $\int_0^T \nabla \cdot f(\gamma(t))dt \leq 0$

Let $f \in C^1(E)$ where E is an open subset of $\mathbb{R^n}$ containing a periodic orbit $\gamma(t)$ of $x'=f(x)$ of period $T$. Then $\gamma(t)$ is not asymptotically stable unless $$\int_0^T ...
5
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0answers
75 views

general solution of the equation $\frac{dy}{dx} =\exp(y/x)$

How can i get the general solution of the equation a) $\frac{dy}{dx} = \exp(y/x)$ b) $\frac{dy}{dx} = \exp(x-y)$ and $y=2$ when $x = 0$ I tried b) first: This is a first-order nonlinear ordinary ...
0
votes
1answer
24 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
2
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0answers
38 views

Power series to solve differential equations?

We can use the formula $$F(x)=e^{λx} [ ρ-λμ-\dfrac{1}{2} λ^2 σ^2 ]^{-1}. (1) $$ to derive an expression for F(x) when f(x) is any integer power $x^n$. Begin by observing that for the ...
5
votes
6answers
83 views

Need explanation for simple differential equation

I can't figure out this really simple linear equation: $$x'=x$$ I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating ...
0
votes
2answers
42 views

Differential equation has solution of form $AV^{β_1 }?$

in this equation $$\dfrac{1}{2}\sigma^2 V^2 F^{''} (V)+(\rho-\delta)VF^{'}(V)-\rho F=0,$$ $\sigma, \rho$ and $\delta$ are constants, and V y F are functions en dollars, why the solution take the ...
0
votes
1answer
21 views

Why does $t^{i\lambda}=cos(\lambda ln(t))+sin(\lambda ln(t))$?

My equation was $$tf''(t)+f'(t)+\frac{\lambda^2}{t}f(t)=0$$ where $\lambda \in \mathbb{R}$. I set $f(t)=t^{m+1}$ and solve from there to obtain $m=\pm i\lambda$. The general solution then is ...
2
votes
1answer
51 views

Nonlinear first order ODE with quadratic in the derivative

This equation shouldn't be so hard, and yet I'm stymied. $$ \left( \frac{dw}{dz} \right )^2 + \alpha \frac{dw}{dz} + w \beta = 0 $$ with $w(0) = w_0>0$ $w(L) = 0$ for some known L and ...
2
votes
3answers
57 views

2D Heat Equation with special initial condition

I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...
0
votes
1answer
19 views

Solving an ODE using variations of parameters and Wronskian theorem.

So I am attempting to solve this differential equation by trying to follow an example that my professor did in class. I am just not too sure about my answer seeing as WolframAlpha gives me this: ...
1
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1answer
20 views

Picard Theorem globally Lipschitz

I understand the proof apart from the last step which I have highlighted in green.
2
votes
1answer
48 views

Finding Fixed Points for Coupled ODE

I have two coupled equations $$\frac{dx}{dt}=\gamma x\left(1 - \frac{\alpha x+\beta y}{N}\right)$$ $$\frac{dy}{dt}=\theta y\left(1 - \frac{\alpha x+\beta y}{N}\right)$$ where $\gamma , \alpha , ...
0
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0answers
26 views

Bernoulli Differential Equation of Second Order

How one can solve a Bernoulli differential equation of second order? i.e., solve the DE \begin{align} \frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} + q \left( x \right)y = g\left( x ...