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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2answers
9 views

Unable to verify solution to difference equation $m_x - 2pqm_{x-2} = p^2 + q^2$

I want to verify that the solution to the difference equation $m_x - 2pqm_{x-2} = p^2 + q^2$ with boundary conditions $m_0 = 0$ $m_1 = 0$ is $$m_x = -\frac{1}{2}(\frac{1}{\sqrt{2pq}} ...
1
vote
0answers
19 views

Numbers of zeros of solution of differential equation

Assume $a>0$ , $b>0$ and there exists a non-zero function $\phi(t)$ such that is the solution of $y''+(a+bcos(2t))y=0$ and on $(-\pi/2,\pi/2)$ has $2n$ zero. prove that $(2n-1)^2\le a+b $ ...
0
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0answers
21 views

Converting a word problem to algebra

This is a forming of an equation, which I haven't been able to get my head around. I have a worked solution to this problem. Question: For $x\in\mathbb{R}^m$ and $\epsilon>0$, show that ...
2
votes
2answers
17 views

Why is the Wronskian of these two functions equal to $\frac{2}{\sqrt{\pi}}$

I can't seem to get the right answer $ f = e^{\frac{y^2}{2}}$ and $g =e^{\frac{y^2}{2}}erf(y)$ where $erf(y) = \frac{2}{\sqrt\pi}\int_{0}^{y}e^{-\alpha^2}d\alpha$ I get $W = \frac{2}{\sqrt\pi} - ...
0
votes
1answer
26 views

Method of solving $w''(y) - (1+y^2)w(y) = y$

I have not done ode's for awhile and have forgotten a lot of stuff. What method am I supposed to use to solve this $w''(y) - (1+y^2)w(y) = y$ (please don't solve it for me). I guess I can use ...
0
votes
1answer
21 views

Behavier of function and its derivatives at infinty

If $\lim_{t\to \infty}(\phi(t))=0$ and $\lim_{t\to \infty}(\phi''(t))=0$ then can we say $$ \lim_{n\to \infty}(\phi'(t))=0$$ Can we have a $\phi(t)$ such that $\lim_{t\to \infty}(\phi(t))=0$ but ...
0
votes
1answer
19 views

Lambert W function identity from differential equation

For constants $v,K$ and a function $C(t)$, can you prove that if : $$ \frac{dc}{dt} = - \frac{vc(t)}{K + c(t)},~\text{with } c(0) = c_0 $$ Then the solution: $$ \left[ K \ln c(t) + c(t) ...
0
votes
1answer
18 views

Building an nth order ODE in Maple (or Matlab)

The question is simple: given a system of ODEs, how can one construct the equivalent nth order ODE in Maple? In my case I have $$ \begin{cases} y''(t)+x'(t)+x(t)=f(t)\\ y''(t)+z''(t)+z'(t)+z(t)=0\\ ...
3
votes
2answers
76 views

How to solve $y'=\frac{x^2+y^2}{x^2-y^2}$?

Solve ODE: $$y'=\frac{x^2+y^2}{x^2-y^2}$$ I do not know where to start! Any suggestions please?
1
vote
1answer
36 views

Solution of $ay''+by'+cy=0$ with positive constants $a,b,c$ satisfies $y(x)\to0$ as $x\to\infty$

Given that $a, b, c$ are positive constants and $y(x)$ is a solution to the differential equation $ay''+by'+cy=0$, show that $\lim\limits_{x \to \infty} y(x) = 0$. I've been able to determine ...
0
votes
2answers
61 views

How to solve $1+y'^2=yy''$? [on hold]

The equation is $1+y'^2=yy''$ I know the answer is $y(t)=a \cosh \dfrac{t}{a}+b$ I need the complete resolution.
0
votes
0answers
17 views

Why is $\langle f(t,x_n),x_n\rangle\le l\lVert x_n\rVert^2$?

Show that; $\langle f(t,x)-f(t,y),x-y\rangle\le l\lVert x-y\rVert^2\implies\langle f(t,x_n),x_n\rangle\le l\lVert x_n\rVert^2$ Why is the inequality above true ? (It is used to show that the ...
0
votes
1answer
18 views

Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
0
votes
0answers
19 views

General Polynomial Solution to an Infinite Differential Equation

To begin, I would just like to say that I don't know too much about upper level mathematics. I'm a curious highschooler (also homeschooled). I would greatly appreciate if someone would help me answer ...
0
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0answers
18 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 ...
1
vote
2answers
51 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
vote
1answer
29 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
17 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 ...
0
votes
2answers
31 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 ...
0
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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 ...
2
votes
0answers
51 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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0answers
15 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
votes
0answers
28 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 + ...
6
votes
3answers
50 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 ...
1
vote
0answers
15 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 ...
0
votes
1answer
25 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
votes
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
votes
0answers
34 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, ...
0
votes
1answer
42 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 $t$.
0
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0answers
15 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 ...
1
vote
0answers
19 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
33 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
votes
1answer
43 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 ...
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 ...
0
votes
1answer
13 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
vote
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 $Y$
1
vote
2answers
24 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
0answers
35 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
votes
0answers
30 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.
0
votes
0answers
24 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 ...
0
votes
0answers
14 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
vote
0answers
18 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 ...
1
vote
0answers
30 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?
3
votes
0answers
46 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} ...
0
votes
0answers
10 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 ...
0
votes
3answers
79 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
votes
3answers
69 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
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
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 ...
1
vote
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 ...