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

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Given a piecewise initial condition, how can the characteristic curve x be sketched when the solution x does not contain u terms?

The charac equation for x: $$\frac{\text{dx}}{\text{d$\tau $}}\text{=2t}$$ The solution x is $$x=t^2+x_0$$ Note that $$\tau=t$$ There is a problem. In order to sketch x, I require some ...
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0answers
14 views

If a system of ordinary equation is consist of elementary function, then would the answer be also elementary?

If a system of ordinary equation is consist of elementary function, then would the answer be also elementary? With elementary function defined as log, e, trigonometric and polynomial function
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1answer
29 views

Solving a system of coupled differential equations

The system is given by: \begin{align} 2x''&=-6x+2y \\ y''&=2x - 2y + 40\sin(3t) \end{align} The textbook did not go more deeply to give the solving technique of these type of problem instead ...
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2answers
32 views

Is $L=\sin^2(t) \frac{d}{dt}$ a linear differential operator?

Consider the differential operator $$L=\sin^2(t) \frac{d}{dt}$$ If it acts on the sum of two functions, $y_1(t)$ and $y_2(t)$, you get $$\begin{align*} L(y_1(t)+y_2(t))&=\sin^2(t) ...
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1answer
17 views

Solving ODES in PDE

The PDE given as: $$t^2u_t-\text{yu}_x+\text{xu}_y\text{=0}$$ The characteristic equations are: $$\frac{\text{dt}}{\text{dt}}=t^2$$ $$\frac{\text{dx}}{\text{dt}}\text{=-y}$$ ...
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1answer
22 views

Manipulating series to find the recursive formula

Ok so I am stuck. I need to get all the $n$'s to $=0$ but I can't reduce my series which has $n=2$ to $0$ because then I will have undone all my work in the first place to get all the $X^n$'s to the ...
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1answer
20 views

How do I express this system of differential equations in polar coordinates?

I'm supposed to express this system of differential equations in polar coordinates. $\begin{cases} \frac{dx}{dt}=\mu x-\omega y-x(x^2+y^2)\\\frac{dy}{dt}=\omega x+\mu y-y(x^2+y^2)\end{cases}$. I'm a ...
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2answers
14 views

Third order non-homogeneous differential equation

I have no idea on how to work this out. I've tried variation of parameters, undetermined coefficients, making it into a system, etc. $$y'''+2y''+5y'+20e^{-x}\cos(2x)=0$$
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1answer
15 views

Index of differential function

Is it valid to say: $$\frac{d}{dy} \left( \frac{du(y)}{dy} \right)^n = \left(\frac{du(y)}{dy}\right)^{n+1}$$ If so, why?
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16 views

What method do we use to find the solution?

Find the solution of the initial and boundary value problem $$u_t(x,t)-u_{xx}(x,t)=0, x>0, t>0, \\ u(x,0)=f(x), x>0,\\ u(0,t)=0, t>0 $$ (The solution should be expressed as an integral ...
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0answers
30 views

What is the exact solution to this PDE?

I'm in a numerical methods class for my senior year of college, and it's been about 3 years since I took diff eq. We have a problem in which we are using numerical methods to approximate the solution ...
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1answer
26 views

Three-Variable Differential Equation Stability

Discuss the stability of the equilibrium points $(1,0,0)$ and $(1,1,0)$ for the system: \begin{align} x' &= y - y^2\\ y' &= z\\ z' &= x - \cos{z} \end{align} I have found the ...
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1answer
18 views

How to describe behavior of population system, given by system of ODEs?

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
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0answers
14 views

Solution space of Linear homogeneous differential equation

The solution space of a L.H.D.E of order n is a vector space spanned by n base vectors, right? So any solution is then a vector of the solution space -> a linear combination of the base vectors. But ...
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0answers
50 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
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0answers
16 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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1answer
29 views

Wave Equation Partial Differential EEquation

Basically I got a simple wave equation with an extra twist. The PDE is $\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L $ with homogeneous boundary condition As usual, ...
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1answer
27 views

An application of Implicit Function Theorem in differential equations?

Let $f$ be a continuous function from $\Bbb R^3 \to \Bbb R$. By a solution of the differential equation $$f(x,y,\dot{y}) = 0$$ We mean a function $y\colon U \subset \Bbb R \to \Bbb R$ where $u$ is an ...
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2answers
49 views

differential equation question $\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}$ [on hold]

how do you solve this ? $$\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}$$ thank you in advance!
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0answers
18 views

Show that the solution of the Cauchy problem $x(t,t_0,x_0)$, $x(t_0)=x_0$ is definite for all $t\geq t_0$. [on hold]

Consider the system: $$x' = A(t)x + b(t)$$ where matrices $A(t)$ and $b(t)$ are only integrable on compact sets of $\mathbb{R}$. Show that the solution of the Cauchy problem $x(t,t_0,x_0)$ is ...
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the Fisher equation has no positive traveling wave solution [on hold]

Use the linearization method to prove that for any $c\in(0,2)$,the Fisher equation$u_t=u_{xx}+u(1-u)$has no positive traveling wave solution $U(x+ct)$ with $U(-\infty)=0$
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Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system [on hold]

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system $$\frac{du_1}{dt}=u_1(b_1-a_{11}u_1-a_{12}u_2)$$ ...
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2answers
57 views

Can somebody please show me the necessary steps to solve this Calculus problem?

I have a homework assignment that asks me to solve the differential equations and it gives me: \begin{align*} xy^2y' & = 2-x\\ y''+4y & = 8x\\ y(1)& =1 \end{align*} Are these three ...
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1answer
25 views

Analytic solution to Poisson equation

I need to find the analytic solution to this equation, in order to compare it with solution I get from using a numerical solution. However, I have not been able to find the solution. I think I can't ...
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0answers
7 views

Central Difference Method

Solve the following using the central difference method: $y(x)= y'+ y + 2x$ where $0 < x < 4$ with $n=4$ subintervals (thus $h=1$). Given that $y(0)=0$ and $y(3)=1$, find $y(1)$. Really ...
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2answers
9 views

Time dependence of velocity from position dependece of velocity

I know dependence of velocity on position $v(x)$ and I wan't to know dependence of velocity on time $v(t)$ I was thinking that using some chain rules or derivative of inverse it would be possible to ...
0
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1answer
21 views

Solve the Initial Value Differential Equations

I split the equation and got y+1 dy = xysinxdx, then I divided the right side by y to get 1 + (1/y) = xsinx dx. I took the integrals of both sides and got y + lny = -xcosx + sinx + c. I don't ...
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14 views

Second order linear ODE and undamped

I am a bit confused with this problem: An object with mass 1 slug is attached to a vertical coil spring of spring constant of 1 pound per foot. After coming to equilibrium, the object is set into ...
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1answer
26 views

quick question with 2nd order linear differential equations

I am solving $y''+4y'+5y=2e^{-2x}cos(x)$ I am working on determining $A$ and $B$ in the particular function. I have the following 2 equations: for the sine part : $-2A+3Ax-3B+Bx=0$ for the ...
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2answers
30 views

Fourier series of complex diff eq

Can I just use Euler's identity to construct the Fourier Series since it is complex? I was personally thinking I could, but I wanted to be doubly sure.
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0answers
12 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 ...
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1answer
26 views

Need help for this case:

I am learning the artificial potential field method for path planning of mobile robot; artificial potential field method has two components: the first one is attractive force and second one is ...
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1answer
27 views

Topological structure/graph from a paper

This question is based off a paper titled "On designing heteroclinic networks from graphs." I'm having a difficult time visualizing something "drawn in 4-dimensions" projected down to a 2-dimensional ...
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1answer
36 views

Help solving differential equations

I would like to know how to classify the following equations: $y''+ 4y'+5y=2e^{-2x}cos(x)$. Is it a second order linear equation?
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1answer
47 views

Can the following nonlinear first order ODE be solved?

I have tried solving this equation from several manners but no luck. Can it be solved? $$\frac{d f}{d t} = A f^2 +g(t)$$ The solution for the homogeneous is (I think; somebody should confirm) ...
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0answers
32 views

Lotka-Volterra Problem From Arnold's Ordinary Differential Equations

Problem 1 of section 2.7 of Arnold's Ordinary Differential Equations book asks to prove that the period of the oscillations in the Lotka-Volterra model tends to infinity as the initial condition ...
4
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1answer
39 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
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0answers
50 views

Why does $\frac{1}{r}\frac{dr}{d\theta} = \cot \psi$?

In the discussion of linear fractional equations in Birkhoff and Rota's Ordinary Differential Equations, the authors assert that if we convert a DE of the form $y' = F\left(\frac{y}{x}\right)$ to ...
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1answer
24 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
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0answers
11 views

Ordinary Differential Equations self-study reference request

I know there are a lot of reference requests for differential equations textbooks but none seem to be what I need. I'm looking for a book I can use for self study that isn't overly complicated and ...
0
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1answer
23 views

solving a partial differential equation

How can I solve the following equation? $$-f_{x}+yf_{xy}+xf_{yy} = c^{'}(x)(-f+yf_{y})$$ where $f=f(x,y)$ is a real function of two variables $x,y$ and $c=c(x)$ is a real function of $x$. I guess ...
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158 views

Does Tom catch Jerry?

Tom has Jerry backed against a wall. Tom is distance 1 away (perpendicularly). At time t=0, Jerry runs along the wall. Tom runs directly towards Jerry. Tom always runs directly towards Jerry. Tom and ...
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1answer
21 views

How to obtain an exact solution to nonlinear second order ODE

I need help in analytically solving this nonlinear second order ODE, $A y(x) + y'(x) \Bigg( B + \frac{C y'(x)}{D y'(x) - y''(x)} \Bigg) = 0$. Any help is appreciated.
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Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
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2answers
48 views

Techniques to solve nonlinear first-order ODEs

I am trying to solve the following nonlinear ODE: $$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$ where $a, b$ are constants and $a>0$. Moreover, you may assume that $b \neq 0$ if necessary. This ...
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1answer
32 views

Differentiation under the integral

Now I have this expression. $\psi(\theta)=\text{log}\int_{-\infty}^{\infty}\exp{\{\Delta\theta-f(\nu)\Delta^2\}}h(\Delta)d\Delta$. The expression of $h(.)$ is not given. So $h(\Delta)$ is some ...
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0answers
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Step Function Section 6.3 [on hold]

Could anyone help me write the function in terms of unit step function? $$ f(t) = \begin{cases} -5, & 0 \leq t < 1\\ 4, & 1 \leq t < 5\\ -3, & t \geq 5 \end{cases}$$
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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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25 views

How can I solve differential equation near point that is not normal

Let we have the following differential equation : $$2z(z+1)w''+z(z+1)w'-w=0$$ By power series near the point $z_0=0$ the problem that the point $z_0$ isn't normal point for this equation , so how can ...
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2answers
55 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...