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
35 views

Continuouty of solution of IVP problem

$ x'=f(t,x) $ , $x(t_0)=x_0$ is a IVP and $f$ is Lipschitz with respect to $t$ and $x$. Show that the solution is continuous with respect to $(t,t_0,x_0)$. I could show continuity with respect to ...
1
vote
2answers
270 views

Stability of trivial solution for DE system with non-constant coefficient matrix

Given the arbitrary linear system of DE's $$x'=A(t)x,$$ with the condition that the spectral bound of $A(t) $ is uniformly bounded by a negative constant, is the trivial solution always stable? ...
0
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5answers
59 views

Find solution of differential equation $y ′ (t)=−y(t)-\frac1{2}(1+e^{-2t})+1$

Could you help me to explain how to find the solution of this equation $$y ′ (t)=−y(t)-\frac1{2}*(1+e^{-2t})+1$$ Given $y(0)=0$ Thank all This is my answer $$y ′ ...
2
votes
1answer
240 views

Fundamental matrix for a linear ODE system with non-constant coefficients

I'm trying to find the fundamental matrix for the following ODE system: $$ y'=\left( \begin{matrix} 3-\sin^2 t&\cos t\\ \sin t+1&\cos t\sin t \end{matrix} \right)y. $$ If one can come up ...
1
vote
1answer
36 views

Differentiation and discrete translation of real functions

Consider the problem of finding differentiable functions $y:\mathbb R\to\mathbb R$ with the property that: $$\begin{equation}y'(x)=y(x+\xi)\end{equation}$$ for some $\xi\in\mathbb R$. Note that when ...
4
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6answers
1k views

ODE introduction textbook

Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an ...
1
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0answers
34 views

Discontinuous Linear Differential Equation

Consider the IVP $$ y' +p(t)y(t) = 0$$ $$y(0) = 1$$ $\forall\,t\in[0,1],\quad p(t)= 2$ $\forall\,t>1,\quad p(t)= 1$ Solve this IVP by first solving for t ∈ [0, 1] and then for t > 1 to obtain ...
2
votes
1answer
127 views

Simple Example of Method of Characteristics

To find the general solution of an equation such as $u_{x} - yu_{y} = 0$, it is clear by the method of characteristics that the characteristic curves satisfy $dx = -\frac{dy}{y}$ and so we get the ...
1
vote
1answer
56 views

Do integration constants depend on initial/boundary conditions?

I am going through an example in my textbook, which solves the boundary value problem $$u_{x} + xu_{y} = x^{2}$$ such that $u(0,y) = y$. The author finds the characteristics $\frac{dy}{dx} = x ...
8
votes
1answer
193 views

$x''= \frac{Ax+B}{Cx+D}$

Might there be a closed-form solution to the second-order differential equation below?$$x''(t)=\frac{Ax+B}{Cx+D}$$ If not, is there any way to get a power series approximation in terms of the ...
0
votes
1answer
82 views

Another solution of the Legendre differential equation

Legendre polynomials are solutions of the Legendre differential equation $${d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0$$ Since the ODE is of second order, it has a ...
1
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0answers
32 views

Linearizing a DDE about a point

I want to show that the delayed logistic equation $$ N'(t)= rN(t) \left[1-\frac{1}{K} \int_0^\infty N(t-u)k(u) \> {du}\right]$$ where $$k(u)= \frac{1}{\tau} \exp\left(\frac{-u}{\tau}\right)$$ ...
5
votes
3answers
912 views

Parabolic shape in Bow (not arrow!)

This is what I am thinking for some days. And I think here are some experts who can answer this question. If I bend any stick made with material that uniform density and its shape is cylindrical ...
2
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0answers
112 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
0
votes
0answers
32 views

derivative of logarithm of function, and derivative of $x$ with respect to $log(x)$

if $y = \mathrm{log}(f(x))$ then the derivative with respect to $x$ is: $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{f^\prime(x)}{f(x)}$ but what if I want to calculate ...
1
vote
2answers
73 views

Explain how to get the right solution of y $dy/dx=y$

When solving the following equation to find y as a function of x: \begin{equation} dy/dx=y \end{equation} First I divide both sides by $y$ and multiply both sides by $dx$: $dy/y=dx$ Then I ...
4
votes
3answers
62 views

ODE Guidance Needed

This isn't from any homework assignment. I'm just trying to find the solution (either explicit or implicit) for this ODE. $$(x+2y)\,dx + (y)\,dy = 0$$ The first obvious step to take is to check if ...
0
votes
1answer
60 views

Is the differential equation $y'''x''+x^2 y'' +x'y'=0$ linear?

Would an equation like this be considered an ordinary linear differential equation (linear in respect to $y$)? $$\frac{d^3y}{dt^3}\dot{}\frac{d^2x}{dt^2} + \frac{d^2y}{dt^2}\dot{}x^2+\frac{dy}{dt} ...
4
votes
1answer
72 views

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

Some observations : 1) Not Bernoulli/homogeneous 2) Not exact How to solve this and in general how to attempt equations that don't have a standard method ? Appreciate any help
0
votes
2answers
127 views

Eulers method for a non-linear boundary value problem.

As part of an assignment, I have been asked to numerically solve the following 2nd-order differential equation. For those wondering, it is a model of groundwater flow through an aquifer beset on both ...
0
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2answers
50 views

Bounded derivative

I am studying a chapter on Lipschitz condition. I am stuck with what it means to say "bounded first derivative". I searched about it in the internet but couldn't find anything. Could someone please ...
3
votes
0answers
31 views

Second order DE question

I am looking for tips for this equation: $ 4xy''+y'+xy'+\frac{3}{2}y=0 $. I am solving with the substitution y=a(x)b(x), but it is getting messy..
1
vote
0answers
61 views

Lyapunov stability in nonlinear system

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ...
1
vote
1answer
46 views

Differential Equation trouble with bessel functions

Here is a differential equation that I encountered while solving a quantum mechanics problem : $$ \frac{d}{dr} \left(r^2 \frac{dR}{dr} \right) - k^2r^2R = l(l+1)R, $$ where $R = R(r) \ \& \ k \ ...
1
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0answers
37 views

A program for draw differential equations

I need to analyze the autonomous equation $ \ddot x +x +\alpha x^2 = 0 $ I need to analyze its flow about its critical points, can you suggest me a program in which I can draw this equation ( $\dot x ...
1
vote
0answers
26 views

Van Der Pol ODE [duplicate]

Hi I'm having troubles linearising and solving for the Van Der Pol equation $\frac{d^2x}{dt^2} - (1-x^2)\frac{dx}{dt}+x =0$ with initial cond $x(0)=1$ and $x'(0)=1$ Any hints or help would be ...
2
votes
1answer
63 views

Let $F = u \nabla u $. Show that div $F = u \nabla^2 u + \nabla u \cdot \nabla u $

just a quick question, I'm just a bit off track as to how to derive this. This is what I have: $$F = u \nabla u = u(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial ...
1
vote
1answer
42 views

Always combine solutions of ode?

I can't think of an ideal example, but for the ode $y'+y/x =4x$ , do we need to solve the homogeneous equation, and the second and sum them? And what would be the second? Thanks. I feel I should make ...
1
vote
3answers
62 views

Find solution of differential equation $y'(t)=-2y(t)+1$

Could you help me explain how to find the solution of the differential equation $$ y'(t)=-2y(t)+1, $$ with $$y(0)=1.$$ I know that the solution is $$y(t)=\frac12 (1+e^{-2t}).$$ How about the IVP ...
0
votes
2answers
95 views

How can I find the fixed points of this differential equation?

The problem is to find the fixed points for the equation: $ \ddot x + x + \alpha x^²= 0 $ (and then sketch the global flow of the equation) (for $\alpha>0$) I know that for the autonomous ...
4
votes
2answers
245 views

On why we have $dy = f'(x)dx$?

I am following Ordinary differential equations by Tenenbaum. Page 48 The differential is defined as: $$dy(x,\Delta x) = f'(x) \Delta x$$ Note: we will want to apply this definition to the function ...
2
votes
0answers
42 views

Nontrivial solutions for a system of equations

Consider $t:[0,1]^2\to R$ that is differentiable a.e. and satisfies conditions (i)-(ii): (i) $$ \int_0^1 \frac{\partial t}{\partial t_1}(x,y)f(y\mid x)\,dy=0, \quad \forall x\in[0,1] \\ \int_0^1 ...
3
votes
1answer
45 views

Solution of $\dot{V} (t)\le -\alpha V(t)-\beta $

Does anybody know the solution of $\dot{V} (t)\le -\alpha V(t)-\beta $ in terms of some inequalities? Where $V(t)\ge 0$ and $\alpha,\beta$ are positive constants; the overdot means the time ...
1
vote
0answers
128 views

Find two linearly independent solutions. Use Frobenius method to solve this equation.

$\displaystyle 2xy'' + (1-2x^2)y' - 4xy = 0$. This is what i got when i solved for the recurrence relation: $(2(n+r)(n+r+1) + (n+r+1) )c_{n+1} =( 2n+2r+2)c_{n-1} $ $\displaystyle c_{n+1} = ...
1
vote
2answers
40 views

help solving 1st order ODE

Here's the equation that's plaguing me: $$xy'-y-x \sin \left(\frac yx\right) =0$$ I've tried u substitution using $y=ux$ and ended up getting: $$\frac 1x\left[\ln \left(\left|\sin \left(\frac ...
2
votes
2answers
42 views

Help figuring out what this textbook did.

I was reading my classical mechanics textbook and this appeared in the chapter for oscillations. $$\dfrac{\mathrm d^2x}{\mathrm dt^2}+\omega_0^2x=0\tag{3.31}$$ We can obtain the equation for the ...
1
vote
1answer
51 views

Relative error in differential equation

Consider the following problem with; \begin{cases} y'(t)=3y(t)-3t & \\ y(0)=\frac13 \end{cases} If the initial value is replaced by $y(0)=\frac13+\epsilon$, compute the relative error of ...
1
vote
1answer
52 views

Existence and uniqueness for 2nd order ODE

I believe I understand the general intuition behind the EUT for 1st order ode's, but why is continuity sufficient and differentiability is not necessary for 2nd order linear odes, but is ...
0
votes
0answers
16 views

Degrees of freedom of differential algebraic equations (DAE)

I have a set of DAEs that seem to be giving the solver difficulty (the solver is APMonitor, a web service), and I suspect I haven't formulated them correctly. The physical system is a pair of rigid ...
0
votes
1answer
14 views

Differentiating functions with 2 variables problem

What I Have: Let $E=y+lnx$ and consider $u=x^2f(E)$ Now I'm guessing I have to differentiate it using chain rule somehow
0
votes
1answer
29 views

Find the co-ordinates of the point on the curve

Calculate the points on the curve $y=(1-x)^4$ at gradient = -4 I solve little bit $\frac{dy}{dx} = 4(1-x)^{4-1}\cdot \frac{d}{dx} (1-x)\\ = 4(1-x)^3 \cdot (-1 )$ the gradient is =-4 so I put ...
1
vote
1answer
30 views

simultaneous diff-eq manipulations

According to my instructor, I should be able to manipulate these differential equations: $\dot{p_1} = -x_1 + p_2$ $\dot{p_2} = -x_2 - p_1 + p_2$ to solve for $p_2$ (not the derivative or integral ...
0
votes
1answer
106 views

How to nondimensionalize the equation $dN/dt=N[s - m(N - a)^2]$?

I have a differential equation for population growth, which is $$dN/dt=N[s - m(N - a)^2]$$ How to nondimensionalize the equation so it can depends on a single dimensionless parameter $k= ...
1
vote
2answers
67 views

How to solve $y^{\left(4\right)}-6y^{\left(3\right)}+9y''=x+\cos3x$?

Could you please give me some hint how to solve this equation? $$y^{\left(4\right)}-6y^{\left(3\right)}+9y''=x+\cos3x$$ I find that the solutions of homogeneous equation are ...
3
votes
5answers
80 views

Linear independence of the functions $1,\cos(x),\cos(2x)$

I want to show that the functions $1,\cos(x),\cos(2x)$ are linearly independent in $C[-\pi,\pi]$. I computed the Wronskian determinat of these functions but at the points $x=0,-\pi,\pi$ the obtained ...
1
vote
1answer
65 views

Analytic solution for the ODE $\frac{d y(x)}{dx}=\frac{a*y(x)^3+b}{c* y(x)^3+ d}$

I have an ODE $$ \frac{d y(x)}{dx}=\frac{a*y(x)^3+b}{c* y(x)^3+ d},$$ where $a,b,c,d$ are constants, and I would love to solve it analytically. I tried Maple 15 and 17 and got $$ y \left( x \right) ...
1
vote
1answer
47 views

sturm liouville system of equations

Consider the following Sturm Liouville system : $y'' + \lambda y=0$ $y(-\pi)=y(\pi)$ $y'(-\pi)=y'(\pi)$ I had to find the eigenvalues and corresponding eigenfunctions for this system , well i ...
1
vote
2answers
36 views

Exact Differential Equation?

I tried to solve this equation so far, since the partial derivative respect to $x$ and $y$ are not exact, I have to find the $u(x)$ to make them exact $e^x \cos(y)dx+\sin(y)dy=0$ Partial ...
3
votes
1answer
46 views

2nd oder odes with non constant coeff.

I am trying to solve these two DE: $ y''+(2x)/(1+x^2)y'+1/(1+x^2)^2y=0 $ and $ xy''-y'-4x^3y=0 $ and I am looking for methods on how to find the sollutions. Should I go with the series method or is ...
2
votes
0answers
50 views

How to classify the equation $\frac{dy}{dx} + x^{2}y = xe^x$

I have the following "homework" problem: Classify each of the following differential equations as ordinary or partial differential equations; state the order of the equation; and determine ...