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

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Solving Systems of ODEs

I am working on the following problems. However, I am unsure how to solve problem (b). So far my attempt is to plug in $x_1 = ake^{kt}$ and $x_2 = bke^{kt}$ so then I get this after row-reduction: ...
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1answer
27 views

ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions

I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be ...
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3answers
52 views

True or False? The differential equation $y'=x^2 \sin y$ has infinitely many exceptional solutions.

I am not sure what "infinitely many exceptional solutions" means exactly. Need help interpreting this and working out a solution if needed. I tried to work out a solution: $$\frac{dy}{dx} = x^2 ...
3
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1answer
43 views

Differential Equations Lectures or books from a theoretical perspective?

I am looking for some differential equation lectures from a theoretical perspective, not a strictly computational one. I found the MIT 18.03 lectures which (as the professor says towards the end of ...
5
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1answer
113 views

Galerkin methods for odes

Could you give me some information about the multi-adaptive Galerkin methods for odes?? What does the term "multi-adaptive" mean?? Are there real-world problems at which we could apply these ...
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0answers
16 views

Quick question about the Frobenius method

When solving the the eigenvalue equation $\mathcal{L}\phi = E\phi$, where $\mathcal{L} = \left \{-\frac{d^2}{dx^2} + x^2 \right \}$ is a Sturm-Liouville operator, using the Frobenius Method $\phi = ...
2
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0answers
37 views

Prove using Green's theorem that the boundary value problem has at most one solution

Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( ...
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3answers
31 views

Trouble Solving a system of linear ODEs

I am trying to solve: $x' = 9y$; $x(0) = -2$ $y' = -16x$; $y(0) = -4$. I am unsure how to even begin. Any help is appreciated.
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0answers
38 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
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3answers
20 views

Solving a system of linear ODEs using variables

I am trying to solve the following system of linear ODEs. $\dfrac{dx}{dt} = x-4y$ $\dfrac{dy}{dt} = 4y$. The initial conditions are $x(0) = -1$ and $y(0) = -3$. I have tried letting: $u_1 = x ...
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0answers
20 views

Solution to non linear ODE $axy''+bxy'+cy+d\frac{y'}{y}=e+fx$

Does anyone know if there is any known solution to the ODE below? $$axy''+bxy'+cy+d\frac{y'}{y}=e+fx$$ where a,b,c,d,e,f are constants. I really want to avoid going numerical, but I could not find ...
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1answer
21 views

Determine the local truncation error of the following method

Consider the ordinary differential equation $$y'(t)=f(t,y(t))$$ Let $y_n$ be an approximate to $y(t_n)$, where $t_n = nh$ and h is constant step size. Determine the local truncation error of the ...
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1answer
42 views

Differential equation of $F(y,y',y'') = 0$ type

I'm studying to solve the deferential equations of $F(y,y',y'') = 0$ type by using the example of the solution for the equation: $$(1 + yy')y'' = (1 + (y')^2)y'$$ In the example one uses the ...
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0answers
38 views

The following ODE global existence theorem reference?

There is an ODE existence theorem of the form: Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function. Suppose that there is a constant $c$ such that if $y$ is a ...
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1answer
16 views

4th order non homogeneous coefficients differential equations

I'm seeking for the solution of the following ODE: \begin{equation} x^3y''''+8 x^2y'''+8 xy''-7y' = 0 \end{equation} I simply interested in the technique...
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1answer
99 views

$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t$ with $x_t$ Ornstein Uhlenbeck process - what to do? [UNRESOLVED]

I consider the following equation: $$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t, \tag{1}$$ where $a=$ constant and where $x_t$ follows an Ornstein Uhlenbeck process (see here under Alternative ...
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0answers
20 views

Hypergeometric differential equation with nonlinearity

I have come across a problem involving a hypergeometric differential equation (http://mathworld.wolfram.com/HypergeometricDifferentialEquation.html) with a nonlinear term added as in ...
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0answers
5 views

Rosenbrock Method Implementation (order 2,3)

for the solution of a (stiff) inhomogeneous 4th-order-ODE I use the Rosenbrock-method which is implemented in matlab (ode23s). I now have the problem that I have to move to implement such a method by ...
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0answers
23 views

Initial and boundary value problem

I have to find the solution of the initial and boundary value problem $$u_{tt}(x,t)-u_{xx}(x,t)=1+\sin x, x \in (0,1), t>0 \\ u(x,0)=\sin x, x \in (0,1) \\ u_t(x,0)=0, x \in (0,1) \\ ...
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0answers
12 views

Using Orthogonal Collocation to solve Coupled Ordinary Differential Equations

I am trying to solve six first order coupled ODE's, two of these are associated with a heat balance of a catalyst pellet, and four are mass balances. I have been trying to solve these equations using ...
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2answers
19 views

General solution of this very simple ODE

I'm looking to confirm that the general solution for $$\frac{\delta ^{2}u(\omega ,t)}{\delta ^{2}t} + c^{2}\omega ^{2}u(\omega, t) = 0$$ is indeed $$A(\omega)e^{c\omega t}\cos(c\omega t) + ...
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0answers
24 views

General solution to a simple ODE in Fourier transform

While going through my notes on Fourier transform, I chanced upon an ODE. The ODE is: $$U_{yy}(\omega , y) - \omega^{2}U(\omega , y) = 0$$ The general solution is: $$ Ae^{\omega y} + Be^{-\omega ...
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0answers
26 views

Steady state response approximation of a linear differential equation using Taylor polynomial

After thinking out how to convert a non-homogeneous linear differential equation, with a polynomial input, to a homogeneous linear differential equation in general for this question I started playing ...
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0answers
13 views

Perform two steps of the forward Euler method for the following system of first order differential equations

$$y'''(t)+sin(t)y''(t)-y'(t)+y(t) = 1$$ $$\frac{dz}{dt} = f(t,z)$$ Converting it to a system of first order differential equations: $$x_1(t)=y(t)$$ $$x_2(t)=y'(t)$$ $$x_3(t)=y''(t)$$ so I can then ...
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1answer
18 views

Convert the equation into an equivalent system of first order differential equations.

$$y'''(t)+sin(t)y''(t)-g(t)y'(t)+g(t)y(t) = f(t)$$ Write the following third order differential equation as an equivalent system of first order ordinary differential equations and write it in the ...
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2answers
37 views

Differential Equations vs Analysis I [closed]

I just got done taking Multivariable Calc and I have room for either Differential Equations or Analysis I for the next summer session. These 15 week courses are condensed into 7 weeks, which one would ...
2
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2answers
42 views

converting a differential equation to polar coordinates

I have the following family of autonomous systems, I'm having trouble with part b): $$x'=x(1-\sqrt {x^2+y^2})-y-\epsilon y$$ $$y'=y(1-\sqrt {x^2+y^2})+x+\epsilon(x+x^2+y^2)$$ a) Convert the system ...
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2answers
41 views

Laplace Transform of derivative squared

I'm trying to solve a problem while I'm studying Control Theory and I came up with a difficult question. $ \mathcal{L}\left[y'(t)^2 \right] $ Basically I need to find the Laplace Transform of this ...
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0answers
20 views

Solve first order BVP on MATLAB

I try to solve this equation on MATLAB $$y^\prime(x)=x+y^2(x)\quad\forall x\in(0,0.9),\quad y(0)=1, y(0.9)=32.725$$ I write two function on matlab, on myODE.m file, ...
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1answer
60 views

Green's function and the disappearing homogeneous term

I was trying to write the solution of an inhomogeneous differential equation $(\partial^2_x+m^2)\phi(x)=\rho(x)$ using the Green function: \begin{equation} (\partial_x^2+m^2)G(x,y)=\delta(x-y). ...
3
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1answer
39 views

Can I lower the order of differential equation in this example?

$$y'''' + 9y'' = 18 (9x+2)$$ Can I make a substitution $$y''=p$$ and solve $$p''+ 9p = 18 (9x+2)$$ Or not ?
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1answer
32 views

Continuation of a vector field on projective space

In the book of V.I. Arnold "Ordinary differential equations" there is a lemma which states that any vector field $F$ on $\mathbb{R}^n$ can be uniquely continued to a smooth vector field $\overline{F}$ ...
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0answers
19 views

General solution to this ODE

Was reading through my notes on Fourier transform but couldn't figure how to find the general solution to $$\frac{\delta U(\omega ,t)}{\delta t} = -k\omega ^{2} U(\omega ,t)$$. I must definitely have ...
2
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1answer
30 views

Why is the general solution of this form?

I found the following in my lecture notes: $$u_t=u_{xx}, x \in \mathbb{R}, t>0 \\ u(x,0)=f(x)$$ $$u(x,t)=X(x)T(t)$$ $$\Rightarrow \frac{T'(t)}{T(t)}=\frac{X''(x)}{X(x)}=-\lambda \in \mathbb{R}$$ ...
1
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1answer
32 views

In an ODE dynamic system, is there a convient way or algorithms for estimating the parameters which make the ODE solution satisfing some constraint?

I have construct a ODE dynamic system like this $$molA(t)==sa$$ $$molB'(t)=sb-db\;molB(t)+\frac{kab\;molA(t)\;molB(t)}{molB(t)+Jab}-\frac{kgb\;molG(t)\;molB(t)}{molB(t)+Jgb} $$ $ molC'(t)=sc-dc\ ...
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0answers
22 views

stability of $ \dot x = (x-1)(y-2), ~~~\dot y=(x-3)(y-2)$

Question: I want to determine the point of equilibrium and the stability (asymptotically stable, stable, or intable) $$ \dot x = (x-1)(y-2), ~~~\dot y=(x-3)(y-2)$$ Attempted solution: So it has to ...
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2answers
48 views

Differential equation of second order (non-linear)

Is there a proper way of solving this differential equation of the second order? $$ \frac{d^2y}{dx^2}=ay^2 $$ Is it even possible?
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0answers
9 views

Finiteness of valuation function defined in Levelt filtration.

I'm studying the Levelt filtration and a certain valuation function comes up and I'm trying to understand when (and why) it is finite. Let $S$ be a disk in the complex plane centred at $0$, $S' = S ...
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1answer
49 views

Differential equation of second order [duplicate]

What is the way of solving differential equation of the type: $$ \frac{d^2y}{dx^2}=ay^2 $$ Is there even a way to do this?
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1answer
17 views

Determining the equilibrium points.

\begin{equation} \frac{du(t)}{dt}=au(t)v(t)-bu(t) \end{equation} \begin{equation} \frac{dv(t)}{dt}=-au(t)v(t) \end{equation} \begin{equation} \frac{dw(t)}{dt}=u(t) \end{equation} \begin{equation} ...
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1answer
59 views

Solving the differential equation $x x''(t)=\frac{1}{t^3-t}$

Solve the following differential equation $$x x''(t)=\frac{1}{t^3-t}$$ I tried to integrate both members: $$x(t) x'(t)-\int [x'(t)]^2 dt=\frac{1}{t}+\log\left|\frac{1}{t}-1\right|$$ but the situation ...
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1answer
15 views

integrating factor for differential equation

$$ \omega_\xi - \frac{\omega}{{\xi}} = \xi e^{\xi} $$ I dont understand how to get the integrating factor for this equation, the answer is $ \frac{1}{\xi} $ how to obtain this? someone please ...
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1answer
22 views

Showing that there is a unique solution for the following equation

Let $I = (0, 1)$, $a : H_0^2(I) \times H_0^2(I) \to \mathbb{R}$ a continuous bilinear form defined by $$a(u, v) = \int\limits_I u'' (x) v'' (x) dx.$$ Show that for every $f \in L^2(I)$ there is a ...
1
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1answer
24 views

Symmetry and periodicity of ODEs solution

Consider a set of smooth ODE: $$(1)~~~~~~~~~\dot{x} = f(x)$$ with $x \in \mathbb{R}^n$ and $f : \mathbb{R}^n \to \mathbb{R}^n$. Consider also a linear transformaton $\gamma : \mathbb{R}^n \to ...
2
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1answer
32 views

Find a quadratic equation that approaches exponential equation.

I have the following equation: \begin{equation} \frac{dy(x)}{dx}=1-y(x)-e^{-ay} \end{equation} I need to find a quadratic equation that approaches the right side of the equation. I know it's ...
2
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1answer
27 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
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1answer
38 views

Simple Laplace equation with peculiar boundary condition

I am looking for a solution to this problem: $$\nabla^2 f = 0$$ $$\lim_{r\rightarrow\infty}f(r,\theta)=1$$ $$f(R,\theta)=\cos(\theta/2)$$ on the domain constituted by the the whole real plane minus ...
1
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1answer
27 views

Cauchy problem \begin{cases} \frac{1}{y'}-x=y x^2, \ \ \ x>0, y>-\frac{1}{x} \\ y(1)=0 \\ \end{cases}

Discuss the following Cauchy problem. \begin{cases} \frac{1}{y'}-x=y x^2, \ \ \ x>0, y>-\frac{1}{x} \\ y(1)=0 \\ \end{cases} My approach: $$y' (x+y x^2)=1\Rightarrow \ y' (1+y x)=\frac{1}{x}$$ ...
3
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0answers
71 views

Differential Equations which involve Infinite Series

The problem statement is as follows: Find the general solution for the following equation for $x(t)$. $$x''+ 9x = 2 + \sum_{n=1}^\infty \cos(nt)/n^3$$ I can't find anything about this in my ...
0
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0answers
30 views

Gronwall's Lemma with an additive term

Suppose that $V(t)$ is a real function of $t\in \mathbb{R}$, differentiable on $t>t_0$, and satisfies $$\dot{V}(t) \le -\alpha V(t) + \beta \sqrt{V(t)}\quad \forall t>t_0,$$ where ...