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

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6 views

Hanging rope - Hyperbolic cosine

Why a rope hanging between two posts forms a hyperbolic cosine? I request a detailed proof please.
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13 views

Can the Heat Equation be Averaged Over a Region?

I am doing a project for my partial differential equations class in which I am motivating the definition of a weak solution. To get started, I assumed that $T$ was a solution to $-\nabla^2 T = ...
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0answers
5 views

Logistic model. Did I set up the differential equation $(1)$ correctly?

Update: I fixed it. The major mistake I made was that originally put $I(t) = \beta\cdot(P-y(t))$ while it of course is supposed to be $I(t) = \beta\cdot y(t)$. NB: I came up with this problem ...
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0answers
11 views

$\Phi(t)=P(t)e^{tR}$ as a fundamental set for $x''(t)=\sin(t)x'(t)$

Problem. Find $2\times2$ matrices $R$ and $P(t)$ such that $R$ is constant, $P(t)$ is periodic, and $\Phi(t)=P(t)e^{tR}$ is a fundamental set of solutions for $x''(t)=\sin(t)x'(t)$. $ $ Attempt at ...
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0answers
8 views

Existence And Uniqueness Theorem Question

(a) Does the existence and uniqueness theorem guarantee the uniqueness of the solution of the initial value problem $dy/dx = 2x(y-2)^\frac{2}{3}, y(1) = 2$ Attempt: NO because $∂/∂y = \frac{4x}{3 ...
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3answers
25 views

How to isolate term in this differential equation?

I need to solve the differential equation $y\frac{dy}{dx} = (x+7)(y^2+6)$ I know that the first step is to isolate both term each side and then integrate... But I can't figure out how to isolate ...
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1answer
34 views

When integrating, can only one term of an equation be integrated or must entire equation be integrated to maintain equality?

Is integration considered a basic operation in the sense you have to do it to all parts of the equation? $y dy - x dx = 0$ Is it valid to do $\int y dy - \int x dx = \int 0$ but invalid to leave out ...
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1answer
33 views

How to put the equation $y'' + ky =0$ into Sturm-Liouville form?

I just wondering how do you put $$y'' + ky =0$$ into Sturm-Liouville form. Reason: I am trying to determine if the equation is Sturm-Liouville on the interval $[-3,4]$.
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1answer
24 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
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3answers
51 views

solve $y'=ay+b$ [duplicate]

I have this differential equation which I want to solve $\displaystyle\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$ I know it is in the form $y'=ay+b$ But how can I solve it ?
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15 views

Initial conditions of a transformed differential equation

I'm not sure if the title is appropriate but this is what I mean. Suppose $$ \frac{dy}{dt} = y(t),~~~ y(t_0) = y_0~~~~~~~~~~~~~(1)$$ and $$ \frac{dx}{dt} = x(t),~~~x(t_0) = x_0 ~~~~~~~~~~~~~~(2)$$ ...
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3answers
66 views

Why before $e^{x}$,the solution was not possible?

we know the important role of exponential function in solving of ordinary differential equation.But the solution can be done by using another function like $10^x$ or $2^x$.The example below shows ...
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0answers
13 views

Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?
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2answers
25 views

Help with first order linear PDE with initial condition

I would like to solve the following pde: $2y\cdot \partial_x u(x,y)-3x\cdot\partial_yu(x,y)=0$ and $u(x,x)=e^{x^2}$ Without the initial condition I got the following result: ...
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0answers
15 views

Solve DE with trasfer function

How would you solve $\frac{\mathrm{d} C}{\mathrm{d} t}=\frac{F}{V}C_{i}-\frac{F}{V}C-kC^{3}$ given $\frac{F}{V}=0.1, k=0.5,$ $\frac{\mathrm{d} C}{\mathrm{d}}=0$ ? I´m supposed to use a transfer ...
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1answer
18 views

Why is it sometimes it seems like you can integrate with respect to x or y and treat the other as a constant, and other times you can't?

I am very confused right now. I thought we can't just do algebra on an ODE to find the solution. The following isn't allowed: $x dx + (y - 2x)dy = 0$ $x dx=-(y-2x)dy$ $\int x dx = \int -y+2x dy$ ...
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2answers
27 views

Getting wrong answer with exact equation with initial condition.

Solve the initial value problem $(4y+2t-5)dt+(6y+4t-1)dy=0, y(-1)=2$ This is an exact equation with $M(t,y)=\frac{\partial f}{\partial y}=4y+2t-5$ and $N(t,y)=\frac{\partial f}{\partial t}=6y+4t-1$ ...
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1answer
43 views

Differential Equation - Water evaporation

Given that a glass of water is filled to its fullest, $10\,cm$ in height, and that after three days the water level is at $9\,cm$ in height. Find when the glass will be empty. The water is ...
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5 views

Locally Linear Systems-repeated $\lambda$

For a locally system whose corresponding linear system has repeated eigenvalues, the type of equilibrium point cannot be determined. I know that the locally Linear system equilibrium can possibly be a ...
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2answers
34 views

How to find eigenvalues of this matrix

How to find eigenvalues of this matrix: $\left( \begin{array}{ c c } 2 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & -1 & 2 \end{array} \right) $ ATTEMPT: $2-λ [(2-λ)(2-λ) ...
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1answer
31 views

Looking for a primitive …

The problem is to find $f$ such that $$f^{\prime}(x)+\int_0^x f(t)\times u(t)dt=0$$ where $u$ is given. I tried to find a primitive of the function $\frac{f^{\prime\prime}}{f}$ but I think it is not ...
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0answers
14 views

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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2answers
31 views

Solve $x''(t)-\frac{x^2(t)}{\sin t}=\frac{\sin\left( (t-1)^2\right)}{\sin t}$.

Solve the following Cauchy problem: $$x''(t)-\frac{x^2(t)}{\sin t}=\frac{\sin\left( (t-1)^2\right)}{\sin t}$$ with $x'(1)=x(1)=0$. I would appreciate some help with this problem. Thank you very ...
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2answers
54 views

solving differential equation second order

Can anyone please explain me step by step how to solve this differential equation: $$\begin{align*} y'' + w^2 y &= 0 \\ y(a) &= A \\ y'(a) &= B \end{align*}$$
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4answers
39 views

Differentiation of function to the power x

Given the function $f(x)=(2 + ln(x))^{x}$, find $f'(1$). This is what I tried: $$f(x)=(2 + \ln(x))^{x}=e^{2x+x\ln(x)}$$ So the derivative would be: $$f'(x)=(2x + ...
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2answers
42 views

Solving system of differential equations $\dot{x}=3x - 2y$, $\dot y = 2x - y + 15 e^t \sqrt{t}$

I am having problem with system \begin{cases} \dot{x}=3x - 2y;\\ \dot y = 2x - y + 15 e^t \sqrt{t}. \end{cases} Eigenvalues are $\lambda_1=\lambda_2=1$, the only eigenvector is $V_1 = (1,1)^T.$ I ...
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2answers
30 views

Differential equation problem. Integrating the logistic equation. [duplicate]

I would like to know how to integrate or rather solve this: $$ \frac{dP}{dt} = kP(L-P). $$ I have the solution, but I would like to know how to arrive at it. I have been told it involves separation ...
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2answers
18 views

Create a non-linear first order differential equation which can be used using the method of separation. [on hold]

In addition, I will need to solve this equation and determine the interval of existence for the solution. Any suggestions?
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1answer
32 views

Solve a second order nonlinear equation

I have a second order nonlinear equation: $$-u''+ \frac{1}{4}(u')^2+au=x^2.$$ I am only interested in the solutions in $[0, \frac{x^2}{a}+\frac{1}{a^2}]$. One paper claims without proof that the ...
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0answers
26 views

Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
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9 views

detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t) $$ where $\eta_a$ and $\eta_m$ are ...
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1answer
39 views

Differential equation $(x^2y^2-1)dy+2xy^3dx=0$

$(x^2y^2-1)dy+2xy^3dx=0$ problem states that $y=t^n$ must be used. Using software it seems that there is a real solution. $$\frac{1}{3} \left(-\frac{\sqrt[3]{3 \sqrt{9 c_1^4-4 c_1^2 x^6}-9 c_1^2+2 ...
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0answers
64 views

Meaning of $dx$ [duplicate]

If I remember correctly, we use $ Δx$ for changes in $x$ and when $Δx \rightarrow 0$ then $ Δx$ takes the form of $dx$?
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0answers
24 views

Solving $y^{(n)}(t)=f(t); t>0$ with initial conditions

I will use the notation $\frac{d^n y}{dt^n} \equiv y^{(n)}$. How do I solve this ODE? $$y^{(n)}(t)=f(t); t>0;\\ y(0)=y_0, y'(0)=y_1, ..., y^{(n-1)}(0)=y_{n-1}$$ What I did: The ODE is in the ...
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0answers
8 views

$0$ is an unstable equilibrium if $f$ is Lipschitz with certain conditions

Consider the following system: $$x'=-x^3-xy^2+2x^2y^2$$ $$y'=-2y+x^2y-3x^3y$$ There are two questions: The first one is to show that $(0,0)$ is uniformly asymptotically stable. The second question ...
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1answer
18 views

intial value piecwise linear ODE; slightly wrong answer

Where am I going wrong? Solve the given initial value problem. Use a graphing utility to graph the continuous function y(x). $\frac{dy}{dx} +2xy=f(x),y(0)=2$ where $f(x)=\left\{ ...
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1answer
84 views

Second order differential equation with a variable coefficient. Show |f(x)| is bounded.

Was given this question as extra credit on an ODE exam. Didn't have time during the exam to consider it, but I have since then, and I'm stumped. $$ f''(x) + f(x) = -f'(x)x^{2015}$$ $f(x)$ is twice ...
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1answer
30 views

Solving ODE for x instead of y

Find the general solution of the ODE. Give the largest interval over which the general solution is defined. Determine any transient terms in the general solution. $y dx - 4(x+y^6)dy = 0$ This is ...
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0answers
9 views

Domain of dependence of wave equation?

Is the solution is $t=R$? Because the domain of dependence of $x=0$ is $|x-0|=t$, so compared to $|x-0|=R$. I get $t=R$. Is that correct? I am not sure if my argument is sufficient. Can anyone help ...
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1answer
19 views

Fundamental solution to the bi-harmonic operator?

I am not sure about what the hint means. If $\Delta u =\frac{1}{2 \pi}(1+\log|x|)$. Since $\log|x|$ is a fundamental solution of $\Delta u =0$. Does that mean $\frac{1}{2 \pi}(1+\log|x|)$ is a ...
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3answers
49 views

How “sharp” does a cusp have to be in order for the equation to be nondifferentiable?

From a mathematical standpoint, I understand the concept of cusps: for example, a cusp exists at the origin of $y=|x|$ because one cannot take the limit from both sides, and therefore the derivative ...
3
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1answer
41 views

Prove the energy is constant in a PDE?

I calculated the $$ \begin{align} \frac{dE(t)}{2\,dt} & = \int_\Omega u_tu_{tt}+DuDu_t+u^3u_t\,dx \\ & =\int_\Omega [u_t(u_{tt}-\Delta u)+u^3u_t] \, dx+\int_{\partial \Omega} u_t ...
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2answers
52 views

Why is it differential equations exist on an interval instead of a domain?

I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of ...
2
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1answer
44 views

Solving $\frac{dy(t)}{dt} = y(t)^2-2 y(t)+2$

How can I solve the following ODE $$ \frac{dy(t)}{dt} = y(t)^2-2 y(t)+2 $$ I'm having a tough time because the differential is in terms of $dt$. My gut instinct is to integrate both sides, but to do, ...
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0answers
6 views

Finding Floquet multipliers of a system of nonlinear differential equations

So I'm wondering how exactly one can calculate the Floquet multipliers of a system of nonlinear differential equations. In this very specific case, the system in question is $$\begin{cases} ...
3
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2answers
42 views

Differential Equation: $\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x$

I'm trying to solve this differential equation and believe I may have solved it using the "separable equations" method. Here's my work: $$\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x = y(x + ...
4
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1answer
47 views

Solve $y' = \frac{1}{2}\sqrt{x} + \sqrt[3]{y}$

Please help with this $$y' = \frac{1}{2}\sqrt{x} + \sqrt[3]{y}$$ Tried making $t=\sqrt[3]{y}$. Then $3t^{2}t'_{x} = \frac{1}{2}x^\frac{1}{2} + t$. $p=t'$. And then expressed $x$ and differentiated ...
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1answer
49 views

a qualitative study of $y'=x(1+{1\over y})$

If I have as initial date $y(0)=\alpha$ for $y'=x(1+{1\over y})$ , the graph of the solution y(x) is under a parable. Can i use the comparison theorem?
3
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2answers
34 views

How to solve linear, second order ODE with Frobenius method with a difficult recurrence relation?

The ODE in question is: $$4xy''+2y'+y=0$$ Shifting the power series of each term so that they are all raised to the power $(n+r)$ will yield this recurrence relation: $$a_{n+1}={a_n\over ...
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1answer
37 views

solving an ODE: problem with integration

I want to solve the ODE \begin{align*} - \left(|u'|^{p-2}u'\right)' & = 1 \quad \mathrm{in}\ (-a,a)\\ u(\pm a) & = 0 \end{align*} for $1<p<\infty$ and $a>0$. I thought I could do ...