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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1answer
24 views

How to solve a nonlinear second order differential equation?

I have been trying to find ways to solve: $$J\frac{d²\theta(t)}{dt²}-K_m cos(\theta(t))=-\tau_f$$ With the initial conditions $$\theta(t=0)=0$$ $$\frac{d\theta}{dt}(t=0)=0$$ Without success. Is that ...
0
votes
1answer
77 views

Examining a solution of a differential equation without knowing the solution

The differential equation is given by $$\dot x=-x \cos x$$ with $x(0)=x_0\in(0,\frac{\pi}{2})$. Now I need to show that for each choice of $x_0$ the domain of the solution $x: I\rightarrow \mathbb{...
2
votes
1answer
81 views

Solve differential equation by using polar coordinates

For $\alpha, \beta>0$ the differential equation, I am trying to solve, is given by $$\begin{pmatrix}\dot x_1\\\dot x_2\end{pmatrix}=\alpha\sin(x_1^2+x_2^2)\begin{pmatrix}x_2\\-x_1\end{pmatrix}+\...
2
votes
2answers
894 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form $\...
2
votes
1answer
49 views

Three coupled differential equations

I am trying to solve the following set of differential equations, I am trying to do it by the usual decoupling methods like adding the equations, subtracting etc which makes the process rather lengthy....
-5
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0answers
18 views
2
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1answer
1k views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
0
votes
0answers
37 views

How to approximate the largest eigenvalue of a monodromy matrix [closed]

Would you happen to know of a method to calculate the largest eigenvalue of a monodromy matrix? For my case the fundamental matrix cannot be calculated explicitly but it exists!
0
votes
1answer
23 views

Boundary of solution of $y'=A(x)y$ for continuous bounded $A$

Given an $a\in\mathbb{R}$ and a continuous, bounded function $A:[0,\infty)\rightarrow\mathbb{R}^n$, I want to show that every solution $\phi:[a,\infty)\mapsto\mathbb{R}^n$ of the ODE $$y'=A(x)y$$ has $...
1
vote
0answers
15 views

Integral and differential inequality

I have integral and differential inequality $y'(t)<Ch^{k+1}+\int_0^ty(s)ds+y(t)$ where $C,h$ are constants and $y$ is positive function with y(0)=0 My goal is to prove $y(t_F)<Ch^{k+1}$ ...
0
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0answers
24 views

Formal approximation for second-order ODE with varying coefficients

I have a differential equation of the form $$0=a+by(x)+cf(x)+z(x)f''(x)$$where the functions $y$ and $z$ are known and we want to find $f$. If $z$ is constant, i.e. $z(x)=Z$, it is straightforward to ...
2
votes
2answers
538 views

Solve Heat Equation using Fourier Transform (non homogeneous)

I know how to solve heat equation where it's like $u_t=k\cdot u_{xx}$ (using Fourier Transform or using Separation of Variables) but this exercise is really difficult for me. I have this: $$u_t(x,t)=...
0
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0answers
26 views

A proof of the test of exactness for differential equations

I went through a proof of the following theorem for test of exactness of differential equations: Let the functions $M(x,y)$, $N(x,y)$, $M_y(x,y)$, and $N_x(x,y)$, be continuous on the region $R=\{(x,...
1
vote
1answer
32 views

Integrating factors, a missing solution

I want to solve the differential equation $(3xy+y^2)+(x^2+xy)y'=0$. If I use the integrating factor $\mu (x)=x$ so that the original differential equation becomes exact, then the general solution that ...
2
votes
4answers
76 views

Find an equation of the curve that passes through the point $(0, 6)$ and whose slope at $(x, y)$ is $\frac{x}{y}$. Book wasn't helpful.

I am using James Stewarts Early Transcendentals Calculus, and Section 9.3 (which is where this problem comes from) doesn't seem to have anything remotely similar to the problem I am facing. No ...
0
votes
3answers
68 views

3rd order differential equation with variable coefficients

How to do I solve this differential equation? $$ x^3 u′′′ + x^2 u′′ + x u′= 0. $$ The series solution method is not working in this case.
1
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0answers
72 views

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$ Method #2

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...
3
votes
1answer
147 views

The ordinary differential equation $\frac{d^2y}{dx^2}-q(x)y = 0$ , $0≤x<∞$ , $y(0)=1 $, $y'(0)=1$ multiple choice question

I am stuck on the following question: Assuming $$\frac{d^2y}{dx^2}-q(x)y = 0,\;\; 0 \le x \lt \infty ,\;\;y(0)=1,\;\;y'(0)=1$$ wherein $q(x)$ is monotonically increasing continuous function,then ...
2
votes
3answers
87 views

Solving the given differential equation.

We need to solve : $$ ( \sqrt{x+y} + \sqrt{x-y}) \,dx + ( \sqrt{x-y} - \sqrt{x+y})\,dy=0$$ I tried as follows : $$ \frac{dy}{dx} = \frac{ \sqrt{x+y} + \sqrt{x-y}}{\sqrt{x-y} - \sqrt{x+y}}$$ And ...
9
votes
1answer
915 views

Damped Harmonic Oscillator and Response Function

This is another one of those questions that I feel like I am almost there, but not quite, and it's the math that gets me. But here goes: For a driven damped harmonic oscillator, show that the full ...
1
vote
1answer
47 views

Can there be a limit cycle without a fixed point in 3D space?

I am working with a population dynamics model. Basically, I have a nonlinear ODE in $R^3$ space, (X,Y,Z), and I know that if I start in the an open region ($0<X<1,0<Y<1,0<Z<1$, ...
0
votes
1answer
28 views

Solving $\sin(\sqrt{\lambda}L) + \beta \cos(\sqrt{\lambda}L)\sqrt{\lambda} = 0$

I'm working with the ODE $$-\frac{d^2u}{dx^2}=\lambda u$$ and trying to find eigenvalues and eigenfunctions corresponding the boundary conditions $$u(0)=0, u(L)+\beta \frac{du}{dx}(L)=0$$ Assuming ...
0
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1answer
102 views

Showing that the omega and the alpha limits are disjoint or have just one common point

Let $f:\mathbb R^2\rightarrow \mathbb R^2$ be a $C^1$ function and $x'=f(x)$. Suppose that there are finites points $x_i\in \mathbb R^2$, such that $f(x_i)=0$. Given $y$, such that $f(y)\neq 0$, and ...
0
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1answer
37 views

Differential Equation using Laplace transformation.

I have a problem solving this differential equation using Laplace transformation. $y'' -9y=0 , \ y(0)=1 , \ y'(0)=0$
1
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2answers
263 views

Laplace transform nonlinear equation

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number? When I say apply the Laplace ...
1
vote
1answer
34 views

Non-linear ODE $\dot{y}=\frac{a}{y}+be^{ct}$ Behaviour for $t\to \infty$ and $t\to 0$

If have to assess the behaviour für small and large $t$ of $$\dot{y}=\frac{a}{y}+be^{-ct}.$$ I know this ODE is a Chini ODE or Abel ODE. I have calculated the implicit solution in Maple (a bit ...
1
vote
2answers
183 views

$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$

Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } x>1\end{cases}$$Then, $...
0
votes
1answer
38 views

Solving a set of two coupled nonlinear ODE's

The equations are: $\ddot{r} - r \ddot{\varphi}^2=0$ and $\ddot{\varphi} + \frac{2}{r}\dot{r}\dot{\varphi}=0$. I unfortunately have no clue how to go about solving this.
0
votes
1answer
37 views

Elliptic limit cycle

The following equations, \begin{equation} \begin{split} \dot{F} &= -C + \frac{F}{\sqrt{F^2 + C^2}}\left(\alpha -\left(\frac{F}{a}\right)^2 - \left(\frac{C}{b}\right)^2\right),\\ \dot{C} &...
0
votes
1answer
31 views

Systems of ODEs

I want to solve a system of ODEs of the following type: $$\large\frac{d\phi_{i}}{dt} = {\mu_{i}}^2\phi_{i} + \sum_{j=1}^{N}a_{ij}\phi_{j}$$ There were IMSL/Visual Numerics routines such as DMOLCH, ...
0
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0answers
52 views

Simple ODE-driven Proof of Euler's Identity

Starting with $f(t) = \cos(t) + j \sin(t)$ and $g(t) = \exp(jt)$, I can show trivially that $f(0) = g(0) (=1)$ and $f' = jf, g' = jg$ I would like to conclude that they are therefore the same ...
0
votes
1answer
22 views

function for which substitution equals expression

I am looking for the function $f(x)$ for which $x=\frac{L}{2}$ results in $\frac{PL^3}{48EI}$. I also know that $f(0) = 0$, $f(L)=0$ and $\frac{df}{dx}_{@ x = \frac{L}{2}}=0$ Do we have enough ...
0
votes
1answer
61 views

Derivative of matrix exponential at $0$

I have to show that the derivative of 'the matrix exponential' $exp: \mathbb{C}^{n\times n}\mapsto\mathbb{C}^{n\times n}$ at the zero matrix $0$ is $id_{C^{n\times n}}$, i.e. $exp(0)=id$. The above ...
1
vote
1answer
31 views

Diffusion equation involving dirac delta term

I've ran across the following diffusion equation: $$\frac{\partial c_i(r,t)}{\partial t}- a \nabla^2c_i(r,t)=b \delta[x-x_1(t)]$$ where $a$ and $b$ are constants related to the context, $\delta$ is ...
1
vote
1answer
68 views

Differential equations: Substitution choice.

This is a rather trivial question, but I am missing something. Consider the differential equation : $$2\,y'+y\,y''=y'\,^2$$ My first attempt was to work along the substitution $z=y'$ and thus the ...
1
vote
1answer
105 views

Coupled differential-integral equation

This is coming from a physics paper I'm reading. It's been a while since I've done much differential equation solving and the system here is a bit unorthodox in that I'm actually searching for the ...
0
votes
1answer
29 views

Sine and cosine solutions of a differential equation

I have to solve a differential equation with constant coefficient such as$$ay'''+by''+cy'+dy=f(x)$$ which has for a characteristic equation$$P_c(\lambda)=a\lambda^3+b\lambda^2+c\lambda+d=0$$First I ...
0
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0answers
34 views

Taking limits of coefficients of differential equation to learn about its solution

I have two questions. Searching online has failed me because searching for "limits" and "differential equations" (DE) virtually always yields results on the behavior of solutions to equations as their ...
0
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0answers
28 views

How to calculate the monodromy matrix of the following ODE system

I have the following equation: $$ \frac{dw}{dt} = (-V(t)+\frac{1}{\lambda}F(t)) w, $$ with $t>0$ and parameter $\lambda>0$ The matrices $F(t)$ and $V(t)$ take the form $$ F(t)= \left( \begin{...
0
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0answers
38 views

Existence of periodic solution of ODE

Study the existence of periodic solutions to the differential equation $$x''(t)+x(t)+\epsilon x(t)^3=0$$ when $0<|\epsilon|\ll1$. I am supposed to use the Poincaré–Bendixson Theorem. This ...
0
votes
1answer
21 views

How to apply boundary conditions correctly?

A short extract from my textbook (which I'm finding a little hard to believe) is as follows: If the general solution to the differential equation of motion of a ...
2
votes
1answer
44 views

Stability of homogeneous linear differential equation with variable coefficients

I would like to know if a homogeneous linear differential equation, with variable coefficients which are periodic, is stable. So the differential equation can be written as, $$ \dot{y}(t)=A(t)y(t), \...
0
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0answers
38 views

What constitutes sensitivity to initial conditions for a system of continuous ODEs?

I am currently working on a disease model (5 dimensions) which seems to exhibit sensitivity to initial conditions. I have used numerical analysis following the method described in Wolf's paper (link ...
1
vote
0answers
52 views

Continuity of the period of solutions of a second order ODE, with respect to their initial conditions

There is a second order ODE $$\ddot{x} + b(x) \dot{x}^2 + c(x) = 0$$ with continuous, and locally lipschitz coefficients b, c : $\mathbb{R}\to\mathbb{R}$. Assume the ODE has 2 partial periodic ...
-3
votes
0answers
63 views

How to calculate the monodromy matrix of the following ODE system (Compensation for answer)

I have the following equation: $$ \frac{dw}{dt} = (-V(t)+\frac{1}{\lambda}F(t)) w, $$ with $t>0$ and parameter $\lambda>0$ The matrices $F(t)$ and $V(t)$ take the form $$ F(t)= \left( \begin{...
0
votes
1answer
39 views

Orbits of orthogonal vector fields in $\mathbb{R}^{2}$ [closed]

Let $f$ and $g$ be two $C^{1}$ vector fields in $\mathbb{R}^{2}$ such that $\langle f(x), g(x) \rangle = 0 \,\,\,\forall \,\, x \in \mathbb{R}^{2}$. If $f$ admits a cyclic orbit, prove that $g$ ...
1
vote
2answers
40 views

How to graph directional field of nonlinear first order ODE and higher order DE's?

For example, if we had this nonlinear first order ODE: $$(y')^2+5yy'-10=0$$ How could we plot the directional field of it ? And what about higher order DE, could we plot directional field to them ? ...
0
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0answers
20 views

Implicit method for ODE

I want to numerically solve the initial value problem of ordinary differential equation for function $u=u(t)$: $$ u'(t)=L(u). $$ I find an second-order implicit method: $$ u_{n+1}=u_n+\Delta t L(u_{n+...
7
votes
1answer
203 views

What is the intuition behind uniqueness of differential equation condition that $f$ and $\frac{\partial f}{\partial y}$ are continuous?

I'm not asking for the proof, I'm just asking for a simple explanation or intuition for this condition. What does it represent ?? Theorem $1.1.$ Let $f$ and $∂f/∂y$ be continuous functions on the ...