Tagged Questions

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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0answers
9 views

Interesting differential equation

Given the continuous function $\mathbf{v}:I\to\mathbb{R}^2$, is it posible to solve the following differential equation: $\mathbf{v}(t)=\mathbf{u}(t)+\dfrac{\mathbf{u'}(t)}{||\mathbf{u'}(t)||}$, ...
0
votes
0answers
20 views

linear first order ODE with polynomial coeffficients

I'm wondering if anything can be said about the solution to a system of linear first order ODEs with polynomial coefficients, especially about analyticity of the solution. The equation is given as ...
3
votes
1answer
24 views

Using y(x_0)=y_0 on differential equation

I don't know what to do next, I'am trying to use the given point but as you can see tan(-pi/2) won't give me the answer that I loking for...
0
votes
0answers
12 views

Splitting Operator

I have a problem with this finite element formulation. After applied a Splitting Operator $Q=\hat{Q} + \tilde{Q}$ I do not know how to procede. I need to obtain the solution of the following finite ...
5
votes
0answers
15 views

Solving 2nd order ODE with Frobenius method - problems with summation symbol

I'm trying to solve the ODE: $$ y''(x) + \frac{2x}{(x-1)(2x-1)} y'(x) - \frac{2}{(x-1)(2x-1)} y(x) = 0 $$ I'm trying to find a solution by the Frobenius method, expanding a power series of the ...
2
votes
1answer
22 views

Periodic solutions of this systems

I need to prove that the system of differential equations $$ \dot x = y \\ \dot y = 1+x^2-(1-x)y $$ doesn't contain periodic solutions. I know the Bendixon criteria (that is to have div no sign ...
-1
votes
1answer
20 views

If a linear ODE system has a solution that tends to zero, it also has an unbounded solution

$a:[0,\infty)\to \mathbb{R}$ is a continous and bounded and $$x'(t)\ =\left(\begin{matrix}0&1\\-a(t)&0\end{matrix}\right) \ x(t)$$ has a non-zero solution like $y(t)$ such that $\lim_{t ...
-1
votes
0answers
18 views

How does one solve ODE $k/\sqrt{x} = y'y''/(1+y'^2)$?

How does one solve the following ODE: $$\frac{K}{\sqrt{x}} = \frac{y'y''}{\sqrt{1+y'^2}}$$ where $y' =dy/dx$?
3
votes
1answer
39 views

show that all other solutions are bounded

Suppose $G(x)$ is a solution of the differential equation $$x'(t)\ =\left(\begin{matrix}-5&2\\-4&1\end{matrix}\right) \ x(t)+ \ f(t)$$ where $f(t)$ is a continous function and ...
0
votes
1answer
36 views

How does one solve $-k(1+(dy/dx)^2)=d^2y/dx^2$?

Suppose that we have the following ODE: $$-k \left(1+\left(\frac{dy}{dx}\right)^2\right)=\frac{d^2y}{dx^2}$$ How does one solve this differential equation to solve $y(x)$? All variables and ...
0
votes
0answers
18 views

Intuitive Understanding of Second Derivative/Concavity.

In Calculus, I understand that derivatives (simply explained) are a rate of change; a slope of a function at a certain point. However, I am struggling to understand the explanations behind the second ...
2
votes
2answers
36 views

How to solve differential equation $dy/dx = y^2/(1+y^2)$ by inegration

My first question is, how does one solve the following differential equation: $$y' = y^2/(1+y^2)$$ My second question is, would it be possible to solve this using ordinary integration method, ...
1
vote
0answers
34 views

Solving for a differential equation

I need assistance in solving this differential equation related to an electronics course: $$V= R\frac{dQ}{dt} + \frac{Q}{C}$$ The solution is supposed to be $$Q = CV \left(1-e^\frac{-t}{RC} \right)$$ ...
1
vote
0answers
18 views

Analyzing differential equations system stability via phase diagram

I´m having a hard time trying to understand how to analyze stability using the phase diagram method for systems, could you please guide me? My result should be just knowing if we´re in front of a ...
0
votes
1answer
31 views

Trying to solve a ODE

I've been trying to solve this ODE, but I'm noy sure if I'm doing right: Solve $(1-xy)\,dx + x(y-x)\,dy = 0$ This equation is not exact, so we calculate its integrating factor $$ \mu = ...
1
vote
1answer
23 views

Solve system of diff equations using laplace transform and evaluate x(1)

I keep getting the wrong answer, and wolphram seems to back me up. Here's the system of equations The answer I get for $x(1)$ is 10492.1... The supposedly right answer is -1426.16 Can anyone try ...
0
votes
2answers
19 views

Solving a differential equation 1st order

The equation is $$y'(t) + 2t^{-1} y(t) = x(t) $$ I keep trying an integrating factor, since it would work out nicely if I could just use $\ln(t^2)$, but it doesn't work out. Help please!
2
votes
1answer
27 views

Solve $x^2(x^2+1)y''-2x^3y'+2(x^2-1)y=0$

Find both solutions to $x^2(x^2+1)y''-2x^3y'+2(x^2-1)y=0$, given that one of the solutions is of the form $y_1 = x^n$ or $e^{ax}$. I can solve $x^2y''+Cxy'+Dy=0$ equations, but dividing by ...
0
votes
1answer
14 views

find specific solution to intial value problem with derivatives

Find a member fo the family that is a solution with conditions $y=c_1e^x+c_2e^{-x}$ domain all real numbers with $y''-y=0$, $y(0)=0$ and $y'(0)=1$ I'm unsure what to do with the derivatives?
2
votes
1answer
23 views

Differential Equation (Advertising Model)

In the sales response to advertising model given by $S'=-(a+ rA(t)/M)S+ rA(t)$, where a,M and r are constants. Assume that $S(0) = S_0$ and that advertising is constant A over a fixed time period T , ...
3
votes
1answer
21 views

Loss of stability (unphysical energy gain) for simple pendulum equation?

I am simulating a pendulum using MATLAB and noted a curious issue. When I use zero velocity and (pi - 0.1) angular position as starting conditions for my second order ODE, the solution deviates from ...
0
votes
4answers
36 views

Solution to $\frac{d^2 y}{dt^2}+y=\sec\left(t\right)$

Is there a solution to the following differential equation? \begin{equation} \frac{d^2 y}{dt^2}+y=\sec\left(t\right) \end{equation}
0
votes
0answers
39 views

Points on a cubic curve

Regarding proof of a converse of a cubic equation's property: A curve passes through 3 collinear points (A, C and E) on x - axis. Points B and D are respectively midpoints of CA and CE. If the ...
0
votes
1answer
21 views

Plotting the phase portrait of $\dot x = x(x-y)$ and $\dot y = y(2x-y)$

I am trying to plot the phase portrait of $\dot x = x(x-y)$ and $\dot y = y(2x-y)$ Now I have already found the fixed points of the system, (0,0). I have also found the Jacobian of (x,y) and when ...
2
votes
0answers
24 views

Black Scholes PDE

How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + ...
2
votes
0answers
31 views

List of ODE's that can be solved by Fourier transform

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier ...
0
votes
1answer
16 views

Is this system of differential equations homogeneous?

I learned that a system of $k$ differential equations is homogeneous if \begin{equation} x_i' = p_{i1}(t)x_i + \cdots + p_{in}(t)x_i + g_i(t), \qquad i = 1, 2, \ldots, k \end{equation} has $g_i(t) = ...
1
vote
0answers
15 views

closed and bounded form

I have this problem, Let $\omega$ a closed $1$ form in $\mathbb{R^{2}}\setminus {0} $ such that $\omega$ restricted to the set $D$ is bounded with $D=\left \{ x\in\mathbb{R} \text{ such that }\left | ...
0
votes
1answer
22 views

Is this equation homogenous or inhomogeous?

I have the following differential equation in my perturbation theory notes $y'' + 2y' = -2y$ $y(0) = 0$ It says in the following section that this equation is inhomogeneous. But I thought ...
1
vote
1answer
13 views

Sum of homogeneous and inhomogeneous solutions also form a solution

For some linear differential operator, $L$, an inhomogeneous differential equation can be formed: $$ L~y(x) = F(x) \text{ with some solution } y_p (x).$$ Similarly a homogeneous equation could be ...
0
votes
2answers
31 views

Greens function method for Newtonian potential

this may be a silly question but, well you know when solving for the Poisson equation that gives the Newtonian potential, $\Phi$, (for a point mass, $M$, at the origin) $$\nabla^2 \Phi = 4\pi G ...
0
votes
1answer
11 views

Finding the max distance between two arrays

I have the solution of an ode as an array $x(t_k)$ where $k=1,...K$. I have another array which is an approximation to the solution of the ...
2
votes
0answers
21 views

Nondimensionization of a simple system.

A damped spring mass system is modelled below: $$m\frac{d^2y}{dt^2}=F_s+F_d\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space t>0$$ ...
1
vote
1answer
21 views

Equivalent of solutions of IVP

Consider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions ...
1
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0answers
11 views

How to solve a kummer equation in term of confluent hyp

How we find the solution of Kummer equation by using confluent hypergeometric function? please do help.
0
votes
0answers
23 views

Finding Convergence Function

$$S(x) = 1+x+\frac{x^2}{2} + \frac{x^3}{1 * 3} + \frac{x^4}{2 * 4} + \cdots + \cdots \frac{x^{2n+1}}{1 * 3*5 \cdots (2n+1)} + \frac{x^{2n+2}}{2*4*6\cdots (2n+2)}$$ How is it to find the convergence ...
1
vote
1answer
20 views

Asymptotic Estimate

Consider the following Sturm–Liouville problem $$u''+\lambda u=0, \ 0<x<1$$ $$u(0)-u'(0)=0, \ u(1)+u'(1)=0.$$ Obtain an asymptotic estimate for large eigenvalues. I solved the problem and ...
1
vote
0answers
26 views

differentiable curve

I´m a little stuck with this problem, I think is false but I can´t find a counter example, here is the problem Let $\omega$ a 1-form defined in $U\subset \mathbb{R^{2}}$(it can be $\mathbb{R^{n}}$, ...
0
votes
1answer
11 views

Using superposition to reduce a complex solution

This is a solution to under-damped harmonic oscillation: $$x = e^{-(\frac{\beta}{2})t}[cos(\gamma t) \pm i sin(\gamma t)]$$ This is the correct reduction according to wolfram (10) $$ x_1 ...
-3
votes
0answers
27 views

Laplace with heaviside step function [on hold]

solve the IVP $$y''-5y'-14y=9t+u_3(t)+4(t-1)u_1(t), \quad y(0)=0, \quad y'(0)=10$$
1
vote
0answers
25 views

Relating Differential geometry with ODEs / conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\phi) := (r(t) \cos( \phi) , r(t) \sin(\phi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
1
vote
1answer
38 views

Solving $ f'(x) =-\log( f(x) +a ) $

Can the solution of $$ f'(x) = -\log( f(x) + a ) $$ with $f(0)=0$ and $a \in (0,1)$ be well approximated by the Lambert W function for $x>0$? It seems that morally this might be the case (by ...
2
votes
1answer
24 views

Expectation and Variance of stochastic equation

My questions is related to this question: Stochastic Differential equation, expectation and variance I.e how do you calculate the variance and expectation of $U_t = e^{-\gamma t}U_0 + \int_0^t ...
2
votes
2answers
40 views

$\frac{d^2 y}{dx^2}-2y=2\tan^3\left(x\right)$

Problem: \begin{equation} \frac{d^2 y}{dx^2}-2y=2\tan^3\left(x\right). \end{equation} using the method of undetermined coefficients or variation of parameters, with ...
1
vote
2answers
67 views

Find all values of $\alpha$ so that all solutions approach $0$ as $x \to \infty$

Given the equation $x^2y′′+\alpha xy′+4y=0$ find all values of α so that all solutions approach zero as $x \to \infty$. Anyone have advice for this question? So I get $y = c_1 ...
2
votes
3answers
39 views

Basis of a Kernel

How would i find the basis of the kernel of the differential operator below $$8y'' + 3y' + 7y$$ We know the equation was homogenous and i believe the basis is two dimensional
-3
votes
2answers
19 views

which of the following can be a differential solution for $\frac{dy}{dt}= -Cy$ [on hold]

which of the following can be a differential solution for $\frac{dy}{dt}= -Cy$ a) $y(t)=2cos(Ct)$ b) $y(t)=5e^{Ct}$ c) $y(t)=5sin(Ct) + 2cos(Ct)$ d) $y(t)=5e^{-Ct}$ e) $y(t)=4sin(Ct)$
0
votes
0answers
26 views

How to normalize a differential equation?

I have a differential equation and I am supposed to normalize it with respect to time and amplitude. I have no idea whatsoever what this means and I was unable to find something online. Any help or ...
0
votes
1answer
19 views

A differential equation question in orthogonal trajectories

$$3x^2 - y^2 = c$$ My solution is; $$6x - 2y{dy \over dx}=0$$ $$ {dy \over dx} = {3x \over y} $$ We have also a slope ${dy \over dx}= {-y \over 3x}$ But here, I guess there is something wrong..
3
votes
2answers
94 views

Direct way to solve $y' = y^2$

My problem is to solve the ODE $y' = y^2$. What I would want to do is to divide by $y^2$ then integrate : it yields $y = \frac{1}{c-x}$ on an interval where $y$ does not vanish, then by continuity it ...