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

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0
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0answers
25 views

how do I resolve equations that are both dependant on each other

I'm working on a project concerning the ideal power equation of aerodynamic bodies seen here: $$P = \frac{1}{2}C A D v^3 + \frac{W^2}{Db^2v}$$ where $P$ = power, $C$ = coefficient of drag, $A$ = ...
0
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0answers
17 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
0
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1answer
22 views

Show that $\frac{dx}{dt}=\frac{1}{14}(15-x)$ given $x=15-12e^{\frac{-t}{14}}$

A biologist is researching the growth of a certain species of hamster. She proposes that the length, $x$cm, of a hamster $t$ days after its birth is given by $x=15-12e^{\frac{-t}{14}}$ Show that ...
1
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1answer
24 views

General solution to differential equation, given a polynomial general solution

I am solving one DE and I have to consider the following: $$(y+ax)^n(y+bx)$$ to come up with a general solution to the following differential equation: $$\frac{dy}{dx} = \frac{10x-4y}{3x-y}$$ I ...
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0answers
33 views

Convergence of $\dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}$

Let $x(t)\ge 0$ obey the following differential equation: $$ \dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}, $$ where $b>0$, $\lambda>0$, $\alpha(t)\in\mathbb{R}$ is both lower- and ...
1
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0answers
21 views

Converse of this theorem about existence of Green's function

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
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0answers
19 views

Finding a Lyapunov function for $u''+u'+\sin u = 0$

Now I need to solve $$\frac{\partial V}{\partial x} = (1 + \frac{\sin x}{y})\frac{\partial V}{\partial y}$$
0
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1answer
19 views

Is it possible that a PDE solved by two different analytical methods with same Initial and boundary values give different results?

I have developed two models of same scenario. Both models involve a PDE which is solved with same Initial and Boundary conditions. In one model it is solved with Laplace transform and in other with ...
0
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0answers
18 views

reformulating a BVP into a system of first order ODEs

I need to convert the following bounded value problem $y'''+3y''+2y'^2-5y^2=1$ with conditions $ y(1)=1, y'(0)=1 $ and $y''(1)=0$ Into a system of first order Initial value problems in order to ...
1
vote
1answer
30 views

Largest interval on which the solution exist of ODE $y'=2(1+y)\sqrt{y}$

If $y$ is the solution of ODE $$y'=2(1+y)\sqrt{y}$$ satisfying $$y(0)=0,y(\pi/2)=1,$$ then the largest interval(to the right of origin) on which the solution exists is $1.[0,3\pi/4)$ $2.[0,\pi)$ ...
0
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1answer
17 views

Proving the interval in which a solution is valid

Question: Verify that both $y_1(x) = 1-x$ and $y_2(x)= \frac{-x^2}{4}$ are solutions of the initial value problem $$\frac{dy}{dx}=\frac{-x+(x^2+4y)^\frac{1}{2}}{2}, \ \ \ y(2)=-1$$ and determine ...
0
votes
1answer
6 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
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0answers
20 views

phase diagram of a 2x2 system of O.D.E.

$$\begin{eqnarray} \dot x&=&a\,x+b\,y,\\ \dot y&=&c\,x+d\,y \end{eqnarray}$$ For repeated eigenvalue case, if b−c is positive, then motion is clockwise; if negative, anticlockwise. ...
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0answers
20 views

Finding the General Solution for a System of Differential Equations with Complex Eigenvalues

I think I might just be having trouble with formatting my answer, because I'm fairly sure my work is right up until this point. The question asks to find the general solution to $$X'= ...
0
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1answer
21 views

two dimensional heat equation

Please I really need some help for this exercise, I can't solve it for any ways... I need to prove the maximum principle for the two dimensional heat equation with zero boundary data. Really I need ...
0
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2answers
12 views

Implementing Adams-Bashforth of order 2 (AB2) algoirthm

Assuming we are given the initial condition for an ODE such that: $$ \begin{cases} x' = f(x,t) \\ x(t_0) = x_0 \end{cases} $$ We are going to solve it numerically using AB2. We know that the ...
0
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0answers
15 views

Euler explicit and semi-implicit

I am given a simple dynamic system with an initial condition: $a(t) = 0.9 - 0.1v(t)$ $v(0) = x(0) = 0$ I want to calculate $x(1)$ with a time step of $\Delta t = 1$ using Euler explicit and semi ...
0
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0answers
29 views

Curve fitting on non-linear ODE data

Background The graph below was generated by a set non-linear ODEs. For those of you who might want to know: It shows the maximum distance achieved by a cylinder when fired at a specified ...
0
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1answer
21 views

solution of second order ODE with non-constant coefficients

What is the general solution of $\frac{d^2y}{dx^2}+P[Q-R\cosh(Sx)]y=0$ where $P,Q,R,S$ are real and positive? I tried transforms but cannot get a solution.
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5answers
148 views

Can we abuse traffic patterns to get home earlier?

I had a heated discussion with my co-worker today, and was wondering if someone here could shed some light on this situation. The post is a bit lengthy, but I wanted to put all my intuition down in ...
1
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0answers
20 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
2
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1answer
17 views

Question about finding eigenvectors for differential equations?

I have a non linear system to analyse and sketch the phase portrait of. At one of the equilibria the Jacobian of the linearised system is given by $$\textbf {J}= \begin{pmatrix} 2 & 7\\ 7/2 ...
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1answer
19 views

uniqueness of solutions in the future

I'm asked to proof this: If we have $$f:\mathbb R^d\rightarrow\mathbb R^d, f \in C^1$$ verifying:$$\ f(0)=0 \ $$ $$\ \langle p,f(p)\rangle \leq0 ,\ \forall p \in \mathbb R^d$$ then the initial value ...
1
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0answers
21 views

A specifc solution to Cauchy-Euler differential equation

Find $\alpha$ such as $y=x^\alpha$ is a solution to the differential equation $$x^2\frac{d^2y}{dx^2}+x(1-x)\frac{dy}{dx}-(x+1)y=0$$ (Oxford (2002), modified) We can derivate and obtaint ...
-4
votes
1answer
29 views

Solve the DE: $\dfrac{ dy}{dx} =\dfrac{y(x+y)}{x(y-x)}$ [closed]

I worked it out and my answer came to be $y^3=x^3+3K$. Is my answer correct since I don't have any answer to this question. Thanks
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0answers
27 views

analyze stability of a system of first order differential equations of different types [closed]

I am working in a mathematical model and I need to analyze the stability of the system of differential equations that define the model, but I don't know how. Here I show a system with the same ...
2
votes
3answers
65 views

2nd degree differential equation

Can someone please tell me how to solve this differential equation? $${d^2y\over dx^2} +y=\tan(x)$$ I am a beginner in ODE and have absolutely no idea how to proceed. Can you also site a reference ...
1
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1answer
27 views

How to find the straight line paths of saddle points for a nonlinear Hamiltonian system?

I have the system $$\dot{x}=y+2xy\\\dot{y}=-x+x^2-y^2$$ Which is Hamiltonian with $$H(x,y)=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Now I want to plot the phase portrait for the system so ...
0
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0answers
24 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
2
votes
3answers
49 views

Numerical solution of an ODE system of equations using RK4

I have given an assignment to find the solution to the ODE system of equations as follow: $$\begin{cases} x_1' = x_1 + x_2 \\ x_2' = -3x_1 -10x_2 + x_2 ^2\end{cases}$$ With initial conditions: ...
-1
votes
0answers
30 views

How to find the radius of convergence given recurrence relation?

Given, $$(1-x^2) \frac{d^2y}{dx^2}-x\frac{dy}{dx}+m^2y=0,$$ solving with power series, $$\sum_{n=0}^{\infty} c_nx^n,$$ then, the following recurrence relation is derived, ...
1
vote
3answers
37 views

Finding Laplace Transform of $te^{-t}$

I started with this integral: $$ \int_{0}^{\infty} e^{-st}\cdot te^{-t}dt$$ = $$\int_{0}^{\infty} te^{-(s+1)t}dt$$ let $dv=e^{-(s+1)t}dt, u=t$ and thus $v=-\frac{1}{s+1}e^{-(s+1)t}dt, du=dt$ ...
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0answers
10 views

Exponential matrix using Laplace transform - reference request [closed]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
1
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2answers
28 views

Find the complete solution of differential equation

I´m having trouble with getting the complete solution for this equation. ${dy\over dx}=-y+2e^{-x}-1$ Somehow I´m not getting the same result as my answerlist or CAS tool. Thanks in advance. I´m ...
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0answers
20 views

Determining the r value for a series solution [closed]

I have trouble answering the 2nd part of the question. For the 1st part, I just simply plugged it into the derivatives and I end up getting $r_{1}=1/2$ and $r_{2}=1/3$. For the 2nd part, I tried ...
0
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1answer
30 views

Value of $f'(3^{1/5})$ from the given differential equation

A function $y=f(x)$ satisfies $$xf'(x)-2f(x)=x^4 f(x)^2$$ and given that $f(1)=-6$ and $x$ belongs to all positive real numbers then prove that $f'(3^{1/5}) =8$ I have tried in this way...... Given ...
-1
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0answers
17 views

Given recurrence relation, how to perform the ratio test? [duplicate]

Given, $$(1-x^2) \frac{d^2y}{dx^2}-x\frac{dy}{dx}+m^2y=0,$$ solving with power series, $$\sum_{n=0}^{\infty} c_nx^n,$$ then, the following recurrence relation is derived, ...
-1
votes
2answers
17 views

How can I convert this second order equation into a first order equation? [closed]

In a previous exercise sheet, we were asked to transform the second order differential equation $$ x'' = -x + \alpha x^{3} $$ Into a first order equation. The solutions have since been released, but ...
0
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1answer
28 views

How does $\text d/\text dx (\ln(1/x) + c) = \text d/\text dx (c -\ln(x))$?

I have to find the derivative of the following expression $\ln\left(\frac 1 x\right) + c$ it appears the answer is $-\frac 1 x$ however i am told it can be the original expression can be simplified ...
0
votes
2answers
23 views

Solving This Differential Equation

I need to solve this differential equation $${{{d^2}y} \over {d{x^2}}} - 2{{dy} \over {dx}} - 35y = - (x + 3)$$ I think I need to try a polynomial of the form $ax + b$ but I can't make progress past ...
0
votes
1answer
18 views

Weak or strong Liapunov function

You are given the system $$\dot{x}=-x-xy^2; \dot{y}=2x^2y-x^2y^3$$ (a) What does the linearization about $x^*=(0,0)$ tell us about the local behavior. So $Df(x,y) = \begin{bmatrix} -1-y^2 ...
2
votes
0answers
30 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
0
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1answer
41 views

Applications of the wave equation

I've recently started to take interest in PDEs and how to solve them, and I'm wondering a bit about real life applications of the wave equation. So far I haven't found anything about practical ...
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0answers
43 views

Solving a specific second order ODE [on hold]

I need some one can help me to solve the following equation : $$z_{tt}-z_{xx}-2z_t = \alpha(t,x)(z_x-z_{tx})$$ where $\alpha(t,x) = \frac{4\epsilon x(1+\epsilon t)}{(1+\epsilon t)^2 - \epsilon^2 ...
-1
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1answer
30 views

Solving first order ODE [on hold]

Anyone can help me to solve the two following ODE ? $\frac{dx}{dt} + \frac{1+\epsilon t - \epsilon x}{1+\epsilon t + \epsilon x}$ = 0 $\frac{dx}{dt} - \frac{1+\epsilon t + \epsilon x}{1+\epsilon t - ...
-2
votes
3answers
40 views

ordinary differential equation project suggestion [closed]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
0
votes
0answers
26 views

Generating Function, finding a closed from solution and sequence [closed]

I am having trouble solving the recurrence relation: $na_n = 3a_{n-1} - 4 a_{n-2} + \frac{8(3^{n-2})}{(n-2)!} $ So far I've used the generating function of A and gotten: $A'(n) = A(z)(3 + 4z) + ...
0
votes
1answer
37 views

eigenvaue of Sturm Liouville problem

Let the limit probem $$ \begin{cases} (P(x) y')' + q(x) y' + \lambda r(x) y=0\\ \alpha_0 y(0)+ \alpha_1 y'(0)\\ \beta_0 y(l) + \beta_1 y'(l) \end{cases} $$ with $\alpha_0^2 + \alpha_1^2 >0$ and ...
1
vote
0answers
26 views

ODE: Solve $\frac{{\rm{d}}~}{{\rm{d}}x}\sqrt{f^2+e^{a x}+b}=f$ for constants, $a,b$ [closed]

I would like to know the solution, $f(x)$, for the following ODE: $\frac{{\rm{d}}~}{{\rm{d}}x}\sqrt{f^2+e^{a x}+b}=f$ for some constant $a,b$.
0
votes
2answers
41 views

differentiate and solve $A = \frac{200}r + 3\pi r^2$

differentiate and solve $A = \frac{200}r + 3\pi r^2$ $A = 200r^{-1} + 3\pi r^2$ $A' = 6\pi r - 200r^{-2}$ $6\pi r - 200r^{-2} = 0$ From here I am not sure how to solve the equation with ...