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

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Derivation of Simple Projectile Motion with Drag

Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $g$, I was able to derive its position vector function: ...
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0answers
9 views

Finding initial conditions for which solutions to IVP are periodic

I have an initial value problem x' = Ax A =$\left[\begin{array}{rrr} 1 &1 &0 &0 \\ 3& -1 &0 &0 \\ 0 &0 &0 &-2 \\ 0 &0 &2 &0 ...
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2answers
25 views

Consider the ODE $y'=2\sqrt{|y|}$ where $y \in \mathbb{R}$

Show that there are many solutions to the ODE $y'=2\sqrt{|y|}$ with initial conditions $y(0)= 0$. Later on in the question, it asks me to find all solutions with initial condition $y(0)=0$, so I ...
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8 views

Suppose that the ODE $x'=f(x)$ on $\mathbb{R}$ is bounded, $|f(x)| \leq M$ for all x

Prove that no solution of the ODE escapes to infinity in finite time. What I've gotten so far is: $x' = \frac{dx}{dt} = f(x)$. And, $-M \leq \frac{dx}{dt} \leq M$. Thus, by integrating, $|x(t)| \leq ...
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0answers
38 views

Study of a system of differential equations

I'm asked to study everything that is possible to know about the sytem$$\begin{cases}x'=x^2-y^2\\y'=2xy\\z'=-z\end{cases}$$ My questions here is, how much can be know about it?, how do I know I ...
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12 views

From fundamental solution to differential equation.

There are various techniques to find the fundamental solutions for a given linear ordinary differential equation (ode). I am interested in reverse engineering; to find a differential equation from a ...
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0answers
13 views

Solution of inhomogenous ODE (4th order)

Hello stackexchangers, I have an inhomogenous ODE in 4th order. This ODE is the constitutive law to describe a material by using the "Wiechert model" (p. 15) which is given by $p_0\sigma + ...
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3answers
40 views

Solving $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y }=1$

I want to solve the differential equation $$x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y }=1$$ with the initial condition $u(1,y)=y.$ I'm very unfamiliar with possible methods to ...
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0answers
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Find surface ode or pde

How to find differential equation of surface connecting/spanning semi-circles with their diameters on x-axis: $ x^2 + y^2 = 1 , x^2 + z^2 = 1 $ with Gauss curvature $K = -1$, and, Mean curvature ...
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1answer
22 views

Generalized Eigenvector for 4x4 matrix

I'm working on Systems of Differential Equations and I'm looking to find the Generalized eigenvector for the following matrix: $\left[\begin{array}{rrrr} 3 &-4 &1 &0 \\ 4& 3 &0 ...
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a differential equation system

I am asking if there is a standard way to solve the system: $$ x^{\prime}(t)=y(t)\times u(t)\\ y^{\prime}(t)=x(t)\times v(t) $$ where u and v are smooth functions. thanks in advance.
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28 views

Help with First Order Differential Equations

Solve the given the two equations: $ xdy + ydx = ydy $ and $ (y^2 + 1)dx +(2xy + 1)dy = 0 $ For the first, I can see that solving this with respect to $ dy/dx $ might be a bit tricky. However, ...
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1answer
36 views

Example of a Differential equation whose solution is not defined for all time $t$ [on hold]

Give an example of a differential equation with its domain $R$ and an initial condition for this equation such the solution is not defined for all time $t$.
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If Omega limit set doesn't have periodic orbit then it has a stationary point. [on hold]

If Omega limit set doesn't have periodic orbit then why it has a stationary point? Can some one explain it? I need some hint.
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0answers
12 views

A Question about fundamental matrix of system $x'=A(t)x$

Assume in linear system $x'=A(t)x$ the coefficient matrix $A(t)$ is a periodic matrix with period $T$ and $A(-t)=-A(t)$ . If $X(t)$ be a fundamental matrix for $x'=A(t)x$ such that $X(0)=I$ then show ...
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0answers
16 views

Prove that if (0,0) is a simple critical point of a quasi linear system , then it is necessarily isolated. [on hold]

Consider the system \begin{cases} \frac{dx}{dt}=a_{1}x+b_{1}y+f(x,y) \\ \frac{dy}{dt}=a_{2}x+b_{2}y+g(x,y) \end{cases} it will be assume that $det \begin{pmatrix}a_{1} & b_{1} \\ a_{2} & ...
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30 views

Lyapunov function

How to do this problem? Find a Lyapunov function for $(0,0)$ in the system: $$x˙=3xy^2−11x^2$$ $$y˙=11x^3−4y^3$$ I know there is no formula for finding Lyapunov functions for a system, so how do I ...
2
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1answer
33 views

Solvability of system of differential equations

Given $a_i:\mathbb{R}^n \to \mathbb{R}$ $(1\leq i \leq n)$, I am trying to find the conditions under which the equations $$ \frac{\partial f}{\partial x^i}=a_i(x_1,...,x_n) $$ $$ f(x_0)=z_0 $$ is ...
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1answer
25 views

Level curves and trajectories.

Consider $f(x,y)=(a(x^2+y),3x^4+3x^2y)\; \{a>0\}$ and the system $(x',y')=f(x,y).$ If $H(x,y)=x^n-y,n\in\mathbb{N}$ find $a,n$ in order to make the level curves of $H$ contain the trajectories ...
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1answer
12 views

Problems with solving a system of differential equations

I have problems solving following differential equations $$y'_1 = 2y_1 - y'_2 + y_2\\ y''_2 = -y_1+y'_2$$ I set $u_1 = y_1, u_2 = y_2, u_3 = y'_1, u_4 = y'_2$ which led me to $u'_1= u_3, u'_2 = u_4$ ...
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2answers
16 views

Find equation of Tangent line at $(4, 1)$ on $5y^3 + x^2 = y + 5x$

Can someone help me find equation of tangent line at $(4, 1)$ on $5y^3 + x^2 = y + 5x$ $Y=f(x)$ I dont know how to isolate the $Y$
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2answers
19 views

The Burger's vortex in 2 Dimension - solving Differential equation

After simplifying the vortex equation, I get to this equation: $$ -\alpha y \partial_y \omega = \alpha \omega + \nu \partial_{yy} \omega $$ where the $\alpha$ and $\nu$ are constant values and ...
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0answers
32 views

Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
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0answers
26 views

Express $y$ in terms of $x$

After solving some differential equation I arrived at $$2ln(y-1) +(y-1)=(x-3)+3 ln(x-3)+c$$ but I can't write $y$ in terms of $x$ to find explicit solution.
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22 views

Green function integration

When I'm trying to find the Green Function of Helmholtz equation for a cube $0≤x,y,z≤L$ $$\nabla^2u+k^2u=\delta(\vec{x}-\vec{x}')$$ where u=0 on the surface. I set to find the green function where ...
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0answers
10 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
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0answers
17 views

A question about fundamental matrix of periodic system $x'=A(t)x$

$X(t)$ is a fundamental matrix of linear differential equation $x'=A(t)x$ where $A(t)$ is a periodic matrix with period $T$ . Show that there exist a non-singular matrix like $C$ such that for ...
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25 views

A-stability of Runge-Kutta methods

I am studying Runge-Kutta methods, but I can't understand why explicit Runge-Kutta methods are not A-stable. Someone can explain it to me?
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0answers
44 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} ...
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0answers
7 views

Find a function to satisfy a necessary condition on a system of pdes

Consider the following set of PDE's $\displaystyle \frac{\partial u}{\partial x}(x,y)=f(x,y,u(x,y))$ $\displaystyle \frac{\partial u}{\partial y}(x,y)=1$ $u(x_0,y_0)=u_0$ Show ...
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3answers
74 views

How to solve a differential equation?

I'm trying to solve the system $$\frac{d^4x}{dt}+4x=0,\quad\frac{d^3x}{dt}+x=0.$$ However, I don't know of any method of tackling such a problem. Can anyone please provide a route to a solution? ...
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3answers
64 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
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1answer
21 views

functions U and L solution of a differential equation

Solving this differential equation with an online calculator: $$-(a z+b) y+(c z+d) y''+cy' = 0$$ I obtain something like: $$y(z)=C_1 \exp\left(\frac{-\sqrt{a}z}{\sqrt{c}}\right) ...
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1answer
34 views

Solution to h'(t) = h(t)^j : Wolfram Alpha mistake?

This should be a quick and easy one. Trying to teach myself a bit more about differential equations, so I put the following equation into Wolfram Alpha: $$h'(t) = h(t)^j$$ It gave me the following ...
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2answers
39 views

Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
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1answer
32 views

How to prove that solution of ODE is even function?

Could you please give me some hint how to prove this statement: If $f(x)$ is solution of $y'=4x^3e^{-|y|}$ then $f(x)$ is even function. It is obvious that $f(x)$ increasing for all $x>0$ and ...
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1answer
17 views

Unable to solve particular solution for non homogeneous second order diff. eq.

The book I'm following jumps many "obvious" steps and sometimes I can't follow up. I have the following non homogenous equation. However I'm unable to find the particular solution since I have so many ...
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1answer
19 views

Laplace transform of a differential equation??

Find unique solution of $y′′ + y = f$ using $y(0) = y′(0) = 0$ and periodic function $f(t) = t$ if $0 \leq t < 2\pi$ Attempted work: $L[y'' + y ] = L[f(t)]$ $L[y''] + L[y] = L[f(t)]$ $s^2 L[y] ...
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1answer
19 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
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2answers
19 views

Finding complete general solution of differential equation with repeated roots (undetermined coefficents)

How do you get a complete general solution for a differential like this? $y^{\prime\prime}+6y^{\prime}+9y=14e^{-3x}$ This is what I have so far for the first part of the problem: $yp=Ce^{-3x}, ...
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1answer
24 views

Exponential of Matirx

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc) to calculate this exponential e^At with A (0 9) (-1 0) I'd ...
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1answer
26 views

I want to find Euler-Lagrange equation for the given functional.

I want to find Euler-Lagrange equation for the following: $$J(u) = \int \left( \frac{\psi(x) u + \dot{u}}{\psi(x)u - \dot{u}} \right)dx, \text{where} \ \psi(x) \ \text{is an explicit function of} \ ...
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0answers
36 views

calculate the second derivative using `ode45`

I have a second order differential equation. I am using ode45 to solve the problem. ode45 converts the equations to the first ...
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1answer
32 views

why are these two power series the same

$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$ How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?
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1answer
44 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(x,t)-2Y_{tx}(x,t)+Y_{tt}(x,t)=0, \quad 0<x<1, t>0$$ $$Y(x,0)=Y_t(x,0)=0, \quad 0<x<1$$ $$Y(0,t)=0, \ Y(t,1)=F(t), ...
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1answer
28 views

What type of differential equation is that?

Good day. I can't understand what type this DE has $ (7x-8y)y'=2x^2-y $ I guess it can't be homogeneous or separable equation. And it seems what it is not a linear equation.. Maybe it is Bernoulli ...
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1answer
25 views

Solving a system of Differential Equations: arbitrary constants

For a research project I am carrying out I am required to solve the system: $\frac{dp}{dt} = -lp $, $ \frac{dc}{dt} = lp - kc $ with initial conditions $p(0) = p_0 $ and $c(0) = 0 $. Here, $p,c$ ...
4
votes
1answer
27 views

Existence and uniqueness of soluctions of $y'=xy^{2/3}$

It is asked to analyze the existance and uniqueness of solutions of the ode at every point $(x_o, y_o)$ $$y' = 3y^{2/3}$$ My attempt: We consider the initial condition $ y(x_o)=y_o$. If we consider ...
0
votes
1answer
29 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
0
votes
1answer
11 views

Differential equations resonance

I've got the question 'Solve (c^2)y ' ' + y = 0, y(1)=1, y'(0)=0. Show that as c->0, the solution does not tend to a limit'. From solving the equation I got the roots as +-(1/c)i, and then using set ...