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

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1
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2answers
37 views

Finding a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$

To find a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$ knowing that $u(x,y)=1$ if $xy=1.$ I thought it may be useful to do the change $v=\log x, w=\frac{1}{2}\log ...
1
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0answers
14 views

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 $\mathbf g=\left\langle0,-g\right\rangle$, I was able ...
0
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0answers
8 views

A basic analysis/O.D.E/perturbation theory question

Consider a system of equations $$x'=f(x,y,\epsilon)$$ $$y'=\epsilon g(x,y,\epsilon)$$ I have seen in the book to claim the following: As $\epsilon -> 0$ the limit is $$x'=f(x,y,0)$$ $$y'=0$$ I ...
0
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0answers
11 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 ...
2
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0answers
40 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 ...
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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0answers
50 views

I am required to solve the boundary value problem $y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$ using the midpoint method.

I am required to solve the boundary value problem $$y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$$ using the midpoint method. In order to get two first order equations I have set $u_1=y\space ...
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0answers
9 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
13 views

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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0answers
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 ...
3
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0answers
15 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 + ...
2
votes
3answers
180 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
6
votes
3answers
43 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 ...
0
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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 ...
0
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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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0answers
17 views

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.
0
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3answers
75 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? ...
0
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0answers
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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0answers
13 views

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.
0
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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 ...
2
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0answers
93 views

Question about solutions of $y''+(w^2+b(t))y=0$ .

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
2
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1answer
35 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 ...
3
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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} ...
1
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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 ...
2
votes
1answer
44 views

Find the indicial equation

How do i find the indicial equation to $x^2y''+4xy'+4y=12-12x^2$ I used the frobenius method but got stuck: $x^2y''+4xy'+4y-12+12x^2=0$ $x^2(y''+12)+4xy'+4y-12=0$ ...
0
votes
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 ...
1
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0answers
18 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} & ...
0
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0answers
31 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 ...
4
votes
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), ...
0
votes
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 $u'_1= u_3, u'_2 = u_4$ ...
1
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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 $Y$
0
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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}, ...
2
votes
0answers
33 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 ...
0
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0answers
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
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 ...
0
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3answers
65 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 $e^ {tA}$ ...
0
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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.
3
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1answer
1k views

ODE solving in Scilab

I have a certain ODE problem which needs to be solved using Scilab. dx(1)/dt=k*x(1)-x(5) dx(2)/dt=k2*x(2)-k1*x(1) dx(3)/dt=k1*[x(2)-x(3)] dx(4)/dt=k1*[x(3)-x(4)] ...
0
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0answers
64 views

Uniqueness theorem in differential equation

Let $g(t) , F(r)$ be measurable functions on $(0,a)$ and assume $g(t)$ be nonnegative and $\int_{0}^{a} (g(t)) dt < \infty $ and $F(r)>0$ , and for every $\delta >0$ we have ...
0
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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( ...
1
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0answers
26 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?
0
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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 ...
2
votes
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 ...
0
votes
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) ...
0
votes
2answers
28 views

solvability condition for differential operator

While reading the research article I came across following derivation, given a self-adjoint operator, \begin{eqnarray} L = \frac{d^2}{dx^2} + f(x) \end{eqnarray} \begin{eqnarray} L\psi_1(x) ...
1
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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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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] ...
0
votes
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 ...