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

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Proof of Gronwall's Inequality

I have a question about the proof of Gronwall's inequality as given in Chicone: Ordinary Differential Equations with Applications. Gronwall: Suppose that $a<b$ and let $\alpha, \phi,$ and $\psi$ ...
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75 views

How do impulsive differential equations work? Can you provide an example?

I have heard of impulsive differential equations being used in some epidemiological models of infectious disease. I haven't heard of them before in my math education, and I was wondering how they ...
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Existence of a solution of a nonlinear ODE

I have to show, that the nonlinear ODE $$u'(t)-2u''(t) u(t)=-1,\quad u(0)=1,\,u'(0)=0$$ has a unique solution $v(t)\in C^2(0,T)$ on any Interval $[0,T]$, $T>0$ and that $$\max_{0\leq t\leq ...
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22 views

Checking the solution to a diffential equation.

Is there a quick way to check that the solution to a diffential equation is correct, I know you can diffentiate it and see if it works but this can take a long time (I want to check my answers in an ...
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1answer
184 views

Finding the Green's function for $y'' + y' = f(x)$

I have this ODE: $$y'' + y' = f(x)$$ with $y(0)=0$ and $y'(1) = 0$. I'm trying to find the Green's function. I multiply through by $G$, integrate over the domain and then use integration by parts to ...
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50 views

Which $n$th order differential equations have $n$ linearly independent solutions?

In these notes (p. 28), it is stated that differential equation $28$ is a second order ordinary differential equation therefore there are two linearly independent solutions. Which is the largest set ...
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101 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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30 views

Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
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146 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
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56 views

Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
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55 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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23 views

Convolution of two equations

Find the convolution of $f(t) = t$ and $g(t) = e^{t}$ $$(f*g)(t)= ?$$ If I am correct, I am able to find the Laplace Transform of each individually, then multiply them together. Let $L(x)$ equal ...
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1answer
36 views

interval of solution of a Linear Ordinary Differential Equation with initial conditions

The equation is $$y' + \frac{2ty}{t^2-4} = \frac{2t}{t^2-4}$$ with $y(0) = 1$ as initial condition. What is the solution and its interval? Using some methods of solution I can come up with $y(t^2 - ...
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41 views

Wronskian Bessel Equations

I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is $$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't ...
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87 views

Using polar form to show that a simple critical point is a spiral point

This is the question in my "homework." I say "homework" because it is not picked up or graded but we are supposed to do it for practice, anyhow here's the question: Given the system ...
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41 views

Laplace transform of multiplication of three terms

Okay, so I have $${f}(t)= t\mathrm{e}^{-2t}\sin 2t.$$ In order to do a Laplace transform, I'm pretty positive I cannot just split it up cause that would basically break the rules of math. I ...
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52 views

Meaning of $ dx \times dy = k $

Does $ dx \times dy = k $ have a mathematical meaning? What about when considering $y = y(x)$?
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54 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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1answer
47 views

Which means adjoint problem of a differential equation?

I wanted to know if anyone can help me with the following problem: Get the adjoint problem (differential equation and boundary conditions) for the problem given by: $$\frac{d^2 u}{dx^2}=f(x)$$ ...
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1answer
148 views

Good Source of Differential Equations Problems with Worked Solutions?

I am looking for a good source of problems for differential equations (first order, second order, laplace, convolution, systems). I find it helpful if the question has a worked solution or at the ...
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24 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
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68 views

Stability of solution to first-order nonlinear differential equation

The problem is to consider $u'(t)+u(t)=\cos(u(t))$ posed as an initial value problem for $t>0$ with initial condition $u(0)=u_0$. The first part asks to show that there is exactly one solution $u$ ...
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44 views

Help Finding Critical Points of a Cubic (with 2 parameters)

I am trying to find bifurcation points in 1 dimension, but am having trouble finding critical points of $x'=\mu x -2x^2-x^3+ \delta$ ( where $x$ is my variable, $\mu$ is a parameter, and $\delta$ ...
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85 views

Bessel Functions Proof

How would I even begin to start proving the following? After looking at Frobenius' method and the Rieman P-equation, I started delving into the derivation of Bessel's/Legendre's functions, and I ...
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41 views

Problem with checking whether $x(t)$ can be a solution of any system of first order homogeneous ODE

I need to find out whether $$x(t) = (3e^t + e^{-t}, e^{2t})$$ can be a solution of the system $$\dot{x} = A x\quad \quad (1)$$, where $A$ is a $2x2$ matrix. I'm not sure of my solution, which is the ...
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66 views

How to solve Cauchy problem?

I'm new to this problem. Here is the question. $$(y+xz)z_x+(x+yz)z_y=z^2-1$$ Find the integral surface which the curves it passes are $y=1$ and $z=x^2$ By Lagrange system i found $u$ and $v$. We ...
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1answer
163 views

Solution to ODE Abel Equation

I aim to find the exact form solution to the this ODE: $$\frac{dS}{dw}S = \frac{a}{w}S^2 + \frac{b}{w}S - c$$ where S is a continuous differentiable function of w, real positive and a, b, c are ...
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56 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
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Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
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206 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
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38 views

defining ODEs recursively in maple

I want to numerically look at a system of ODEs with a large number of variables; defined by $da_j(t)/dt= 2^j a_{j-1}^2 - 2^{j+1} a_j a_{j+1}$, for $j=0\ldots50$ with $a_{-1}= a_{51}=0$. In maple, I ...
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3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
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1st order differential equation

I am given the following: $$ \begin{cases} x \ln x \frac{dy}{dx}+y + x = 0, &\mbox{if}\quad x>1, \\ y = 0, &\mbox{if} \quad x=e \end{cases} $$ I tried to separate it and got this: $$ -y \ ...
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2answers
65 views

How to solve $y''+9y=-18\sin{3x}-18e^{3x}$?

Here is my solution so far: $$y''+9y=-18\sin{3x}-18e^{3x}$$ 1.Find complementary soultion.$$y''+9y=0$$ assuming that solution will be in form $e^{kx}$, substitute $y=e^{kx}$, $$k^2e^{kx}+9e^{kx}=0$$ ...
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239 views
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Solve by separating variables

$$\frac{dy}{dt}=e^y +1$$ I've tried: $$dy/dt - e^y = 1 $$ $$\Leftrightarrow y' - e^y dt = 1 dt$$ But I'm not sure what to do next or if I'm even doing this right!
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49 views

An ODE with trigonometric coefficients

Anyone knows how to solve the following equation: $\cos(x) V(x) + \sin(x) V'(x) - V''(x) = 0$ with an arbitrary initial condition, let's say $V(0)= 1$. Thanks ;)
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34 views

Solving first order differential equation

I am given this: $$(2x+1)\frac{dy}{dx}+y = 0$$ I tried this: $$\frac{1}{(2x+1)} dx = \frac{-1}{y} dy$$ Then integrated the above sum and got this: $$ \frac{ln(2x+1)}{2}= -ln(y)$$ The answer is: ...
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160 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
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1answer
19 views

How would you compute eigenvectors from this linear system?

I am stuck on a problem and I do not know how to obtain the eigenvectors: $\frac{dY}{dt}=\bigl(\begin{smallmatrix} -2&0\\ -3&1 \end{smallmatrix} \bigr)Y$ Work: I obtained the eigenvalues ...
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1answer
20 views

Find the function $f(x)$ when it satisfies the ode

The Fourier transform of the function $\frac{d}{dx}f(x)+xf(x)$ is $i[\frac{d}{dk}\widetilde{f}(k)+k\widetilde{f}(k)]$, so if a function $f(x)$ satisfies the ode $\frac{d}{dx}f(x)+xf(x)=0$, then the ...
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54 views

What is the best approach to solve $ 4y^3 y''=16 y^4 -1$?

How can I solve this DE: $$ 4y^3 y''=16 y^4 -1$$ I really would not bother asking if Wolfram alpha had not exceeded comp. time and not shown me step-by-step solution.
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36 views

Method of Undetermined Coefficients: Solving for a particular solution

Solve for $2y''+3y'+y=t$ using method of undetermined coefficients. So I let $Y=At+b$ to solve for the particular solution. After substituting the first and second derivative into $2y''+3y'+y=t$, I ...
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79 views

solve the differential equation y'+sin(x)y=(sinx)^3, y(0)=-3 is the initial condition

I have a problem with the following differential equation, $y'+sin(x)y=(sin(x))^3$ First I have determined that the answer is something like: $y(x)=c(x)* e^{cos(x)}$, but now I have to determine ...
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26 views

How to solve $xy'=2\sqrt{x^2+y^2}+y$?

How to solve: $$xy'=2\sqrt{x^2+y^2}+y$$ And what would be the standard form to illustrate this situation? (e.g. $y' +P(x)y=Q(x)$ would be standard form of first order linear differential equation)
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How can i find the next approximate value with this iteration formula?

My ODE is : $$y'' + 2t(y')^2 = 0 $$ with initial values $$y(0)=2,y'(0)=1$$ and the analytical solution is $$y(t)=\tan^{-1}(t)+2 $$ which we convert to a system $$y_1' = y_2 \\ y_2'=-2t(y_2)^2$$ ...
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97 views

Analysis of stability of a linearized ODE with a periodic solution

I am asked to find the stability of the following ODE: \begin{equation*} \dot{y} = y^{2} + 2\cos(t)\sin(t) - \sin^{4}(t) \end{equation*} by linearizing around a particular solution $\eta = ...
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3answers
116 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
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Lipschitz continuity in two variables [duplicate]

Prove that $y \mapsto f(x,y)$ is Lipschitz continuous, where $$f(x,y) = \frac{y}{x} \ln{\frac{y}{x}}, \ \ \ |x-1| \leq \frac{1}{2}, |y-1| \leq \frac{1}{2}e$$ I tried to solve this, but I find it ...
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53 views

How can I solve an ODE when $F(x_0)=F'(x_0)=0$ is given at an unknown point $x=x_0$ using bvp5c?

I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an ...