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

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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.
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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 ...
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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 ...
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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 ...
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0answers
27 views

Nonlinear differential equation of first order with polynomial terms

I need to solve the differential equation: $(3x^2+8xy^2+(x^3+8x^2y+12y^2)\frac{dx}{dy})=0$ with the condition $y(2)=1$ I'm tried to solve and every time I go in: $$\int ...
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1answer
26 views

Discuss the following graphs(Differential Equations)

So I have a differential equations midterm coming up soon, and in my last exam I messed the graphing question up. It was very similar to the one I am posting. All the questions said was "Discuss the ...
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0answers
43 views

Differential Equation: $f''=f$

I don't know how to approach this problem. So I know that $f=f'$ $=>$ $(log(|f'|))'$ = 1. But, I don't see how that can be used to solve my problem.
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1answer
43 views

A real matrix whose eigenvalues have all negative real parts

While taking a look in some lecture notes of an ODE course, I found the following claim, which appeared in the text as an exercise: Let $A$ be a real $n\times n$ matrix whose eigenvalues have all ...
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22 views

I have a differentiation problem

$x$ and $y$ are unknown variables. $x_i$ and $y_i$ are constant $(i=1,2)$ $N=2$ is here. $\theta=\begin{bmatrix} x & y\end{bmatrix}$. My question: how can I differentiate this equation?
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Hints on solution to $u_t-\Delta u+cu=f$

Consider the problem (Evans, Ch 2, 14) $$ u_t-\Delta u+cu=f ,x \in \mathbb R^n\times (0,\infty)$$ $$ u=g , \mathbb R^n\times {t=0} $$ If $u$ solves $ u_t-\Delta u=f$, $u=0$ on and $v$ solves ...
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Find at least the first four nonzero terms in a power series expension of the general solution about xo= 0

$x^2y''-y'+y = 0$ about $x_0 = 2$ I tried to get an answer by letting $t = x-2$ and assuming $$y = \sum_{n=0}^\infty a_nt^n$$ $$y' = \sum_{n=1}^\infty na_nt^{n-1}$$ $$y'' = \sum_{n=2}^\infty ...
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1answer
21 views

Maximizing dy/dx in a Differential Equation.

Consider the IVP $$ y' = f(t,y) = t^2 + y^2$$ $$y(0) = 0$$ Let R be the Rectangle defined as $$t\in[-1/2,1/2]$$ $$y\in[-1/7,1/7]$$ Find Max f(t,y) where t,y belong to R and find the Lipschitz ...
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0answers
27 views

Steady-state is undetermined in 3 differential equations system?

I have a system of 3 differential equations as: $\dot{x}=(x+2)\dot{y}+\dot{z}$ $\dot{y}=y-3$ $\dot{z}=3x+z-5$ I am trying to conduct a stability analysis around the steady-state. But as you ...
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0answers
17 views

Solution of differential system

Consider the following differential system in $\mathbb{R}^{n}$: $$u'=Au+(x^{2}+1)^{-1}u---(1)$$ where $A$ is an $n\times n$ matrix with real entries such that all eigenvalues of $A$ have ...
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0answers
14 views

Solution of differential equation with imaginary roots

I am stuck with the solution of the following differential equation $y''''-a^2 y=0$ for $a>0$ I think I should solve by using imaginary roots, but this equation got me. Could someone please ...
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1answer
37 views

Doomsday Vs. Extinction - Population Growth Model

So the question goes something like this. Let $P(t)$ be the bunny population in a certain area after $t$ months. If $P(0)=200$, solve this IVP and show all the steps. Find $P(t)$ explicitly and draw ...
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1answer
23 views

Find for which r this system converges to a fixed point

Given the following (discrete time) system $x(k+1)=r-rx(k)$ where $ r>=0 $ is a parameter Find for which $r>=0$ all solutions of this system converge to a fixed point Verify if there exist ...
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1answer
26 views

Solve cauchy problem $y'=1+y^2, p(\pi)=1$

Solve cauchy problem $y'=1+y^2, p(\pi)=1$ I tried to do it in traditional way and my answer was: $y=tg t+C, C=1$ but it isn't correct.
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2answers
52 views

When all solutions of $y''+ay'+by=0$ are bounded in R?

Could you please help me solve this problem. Suppose $y''+ay'+by=0$ is differential equation with $a,b$ are real numbers. I need to find conditions when all solutions of this equation are bounded. I ...
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1answer
13 views

Confusion in finding order and degree of ODE

in this question im asked to find order and degree of ode (y')^2 +5 (y)^1/3 = x confusion is that i can see order is 1 and degree is 2 ... but if i write eqn as (y')^2 - x = 5 (y^1/3) . then cubing ...
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2answers
38 views

$V dV = \frac{1}{2} d(V^2)$?

I'm following a derivation of the Bernoulli Equation for fluid flows from a book and at one point it says - "Noting that $V dV = \frac{1}{2} d(V^2)$"... How is this derived? Here is what I would do ...
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1answer
19 views

To check whether given equation is LDE and to find its solution

consider differential equation |y'| + |y| = 0 satisfying 0 < x <1 and y(0) = 1 im asked to CHECK WHETHER IT IS LINEAR DIFFERENTIAL EQUATION AND whether it has unique solution . Problem is ...
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3answers
796 views

A function equal to its integration?

It is asked that I find a function such that $$10-f(x)=2\int_0^xf(t)dt.$$ I tried giving a new function F(x) such that ${dF(x)\over dx}=f(x)$, but all I got was a new equation ...
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0answers
27 views

Finding if a differential equation has a solution

Here is another problem which i dont have solution as well as idea Consider the differential equation $$ y"(x) + P(x)y'(x) + Q(x)y(x) = 0 $$ then set of initial conditions for which above ...
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2answers
82 views

Difficulty understanding the solutions to $x'' = -\omega^2 x$

For some reasons involving physics, I'm supposed to consider the equation $x'' = -\omega^2 x$. Normally, I would say the solutions are of the form $x = A \cos(\omega t + \phi)$. But when $\omega = ...
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0answers
16 views

Uniqueness Theorem for Second Degree Inhomogeneous DE with Continuous Coefficients

I've been working out of Garrett Birkhoff, Gian-Carlo Rota Ordinary Differential Equations and supplementing with MIT OpenCourseWare Diff. Eqs. courses, and I came upon this problem in the proof of ...
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1answer
36 views

Determining the Euler-Lagrange equations for a minimizataion problem

I'm working on a problem in computer vision and I've ended up trying to minimize the functional $$\int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$ where $\lambda$ is ...
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3answers
44 views

Integration in question could not be resolved.

I do not know how to solve this integration
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1answer
40 views

Some Doubts in ordinary differential equations viz Linear dependence and order/degree Q [closed]

I was doing my homework assignment and encountered few problems and confusions ehere they are 1 . If wronskian of two functions is zero at some point in (a,b) then does it implies that they are ...
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1answer
25 views

Deriving a high ordered Euler-Lagrange equation.

I've been able to derive the Euler-Lagrange equation for $$\int_a^b F(x,y,y')dx$$ relatively easily by using the total derivative and integration by parts. However, I was unable to apply the same ...
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1answer
32 views

Is it possible to separate two variables?

Here is the following problem: $$ (\frac{du}{dv})^2=u(v-u)$$ Is it possible to separate these two variables?
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Nonconstant initial conditions? (ODE)

My prof gave us a quick ODE review question: it's a linear, second order ODE with constant coeffiecients, so should be easy peasy. BUT the ICs are: $f(0)=0$ $f'(0)=-f$ How do I deal ...
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0answers
25 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
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2answers
37 views

Show Laplace operator is rotationally invariant

I'm trying to show the Laplace operator is rotationally invariant. Essentially this boils down to showing $$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 ...
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1answer
22 views

Differential equation of inclined plane

I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$ which appears in page 4 of ...
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2answers
25 views

Can one solve this differential equation without prior knowledge?

Simple question; how can one prove that the solution to $y'^2 = 1 - y^2$ is $y = \pm\sin(x+c), \ c \in \Bbb{R}$, without prior knowledge of the solution?
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1answer
23 views

Euler's formula

How do I used Euler's formula to for $e^i\theta$ to derive the trig functions for $cos(\theta + \omega)$ and $sin(\theta+\omega)$? Do I need to add something to the exponent of $e$ in order for this ...
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1answer
23 views

Solving the ODE $w''(r) + \frac{1}{r}w'(r) - \frac{n^2}{r^2}w(r) = \frac{4\sqrt{r}}{n^2}(-1)^n $

How can we solve the ODE: $$w''(r) + \frac{1}{r}w'(r) - \frac{n^2}{r^2}w(r) = \frac{4\sqrt{r}}{n^2}(-1)^n $$ I think the homogenous equation is of Euler type, and making the standard trial ...
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2answers
32 views

Solution of nonlinear second order ODE is global

Let $y:[0,\infty]\to \mathbb R$ be a solution to: $$\begin{aligned}y''(t) &= -\sin(y(t)^2)\text{ for } t>0\\y(0) &= y_0\\y'(0)&=y_1 \end{aligned} $$ Now I'd like to show that the ...
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1answer
32 views

Solving ODE $\alpha_0''(r) +1/r\space \alpha_0'(r) = -\frac{4\sqrt{r}}{3}\pi^2$

How would one go about solving the ODE $$\alpha''(r) +\frac{ \alpha'(r)}{r} = -\frac{4\sqrt{r}}{3}\pi^2$$ I know that one should use integrating factors, but I always get confused about them - when ...
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0answers
24 views

Mixing Problem, Different Flow Rates

This is a double question and the first part reads Consider a tank holding 100 gallons of water in which are dissolved 50 pounds of salt. Suppose that 2 gallons of brine, each containing 3 pounds of ...
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3answers
26 views

2nd order odes how do I do it

Why can I just change $y''$ to $m^2$? So for example: $$y''+y'-2y=0$$ $$m^2+ m -2=0$$ $$=(m+2)(m-1)$$ $m=-2,1$ $Ae^{-2x}+Be^x$ But where does that change come from? $y''=m^2$ $y'=2m$ $y=2$ Is ...
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0answers
26 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
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2answers
63 views

Separation of variables PDEs

In this answer, he has three cases $(\lambda = 0, \lambda \lt 0, \lambda \gt 0)$. I understand the first does imply it is linear, hence it isn't consistent with the initial conditions, and looking at ...
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0answers
21 views

basic Stochastic differential equation

I'm sorry but I'm having some troubles to find a solution of this simple stochastic differential equation, $dX_{t}=2\sqrt{X_{t}}dB_{t}+2dt$ where $B_{t}$ is a Brownian motion, please can you help ...
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2answers
44 views

Find the functions $f$ that satisfy the given initial value problems

(a) $f'(x)+3x-2=0$, $f(2)=0$ (b) $2f'(x)-\sqrt{x^3} = 0$, $f(0) = 3$ I know the functions need to be integrated to find $f(x)$, however I am unsure as to how to integrate $f'(x)$ in the ...
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1answer
38 views

Solving Initial Value Problems 2nd Order DEs

$$3(y-1)^2 = y''$$ where $y(0) = 3$, $y'(0) = 4$ and we assume $y \neq 1$. I know how to solve IVPs in when the 2nd order DE is in the form: $$p(x)y'' + q(x)y' + g(x)y = 0,$$ however, I think this ...
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1answer
16 views

Second Order Ordinary Differential Equation with Unknown Scalar y(t)

I'm given $\frac{d^2y}{dt^2}(t) + (\sqrt{1+y^2}-2)\frac{dy}{dt}(t) + y(t) = 0$, and that $y(t)$ is a solution to the above, constant such that $y(t) = c$. I need to solve for $c$ but I'm not sure ...
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3answers
58 views

How to solve a differential equation that is equal to a constant?

How to solve a differential equation: $${{d ^2 u} \over {d x^2}} + u = k,$$ where $k$ is some constant number? I know that if this was ${{d ^2 u} \over {d x^2}} + u = 0$, then an auxiliary equation: ...
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1answer
24 views

Finding parameter for ODE to get polynomial solution

The ODE $$ (x^2+2x)y'' + \theta y=0 $$ has a quadratic polynomial solution. I want to find $\theta$ for which the solution is a quadratic polynomial. I assumed solution in the form of $y=ax^2+bx+c$ ...