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

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What is the Fourier Transform of this function $ \frac{e^ \frac{1}{2}t(-w^2 +2 \sqrt{2 \pi} u \delta''(w)) }{ \sqrt{2 \pi}} $?

As the title says, what is the Inverse Fourier Transform of this function: $$ \frac{e^ \frac{1}{2}t(-w^2 +2 \sqrt{2 \pi} u \delta''(w)) }{ \sqrt{2 \pi}} $$ The inverse should be taken with ...
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37 views

What is the general solution of this differential equation? (with full steps please!) [on hold]

I was wondering guys if anyone can help me get the general solution of this equation. In fact, I haven't studied general solutions at all so please write steps as much as you want. $$ y = C\sqrt{1+(...
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24 views

Using LaPlace transform in ODE

I am looking to use the Laplace transform to solve the following equation: $ y''+16y=(t-3)u_3(t), y(0)=0, y'(0)=1 $ My solution: So I see the left side of the equation works out to be: $ s^2Y(s)-...
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33 views

Partial Differential equation problem with separation of variables

(A) Assuming that $u(r,t)=R(r)T(t)$ is a solution to the partial differential equation \begin{equation} \dfrac{\partial^2u}{\partial r^2}+\frac{2}{r}\cdot\dfrac{\partial u}{\partial r}=\frac{1}{k}\...
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2answers
17 views

A question in Diffferential Equations

Looking to express this function in terms of the unit step function. $$ f(t) = \begin{cases} t^2, & 0\le t<2, \\ 1, & t\ge 2. \end{cases} $$ My solution: $$ f(t)= t^2[u(t-1)-u(t-2)] $$ ...
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2answers
48 views

Solve $y''+(y')^2+1=0$ as a separable D.E.

Solve the equation $y''+(y')^2+1=0$ . I came across this problem in an old textbook of mine under the section of Separable Differential Equations. I am just not seeing how to do it! I'm not looking ...
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2answers
63 views

Why are Lyapunov functions always quadratic?

Consider stable linear system $\dot x= Ax + Bu$. We’ll show that the Lyapunov bound is tight with $V (z) = z^T W^{−1}z$. Multiply $AW_c + W_c A^T + B B^T = 0$ on left & right by ${W_c}^{−1}$ to ...
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3answers
27 views

Finding a differential equation orthogonal to a family of curves

The question is: Consider the family $F$ of circles in the $xy$ plane, $(x-c)^2+y^2=c^2$ tangent to the $y$ axis at the origin. Find a differential equation that is satisfied by the family of curves ...
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Using symmetry to solve Non-Linear Ordinary Differential Equations

I know that general rules or general guidance for solving nonlinear differential equations do not exist, but im curious about the various ingenious ways that are being used to solve some of them. I ...
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1answer
15 views

specific solution the the Burger's equation $u_t + u u_x =0$ with boundary condition $u(x,0)=e^{-x^2}$

I have difficulty with finding specific solution to the below PDE $$\left\{\begin{matrix} u_{t}+uu_{x}=0\\ u(x,0)=e^{-x^2} \end{matrix}\right.$$ My attempt: It is stright forward to get the ...
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1answer
1k views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
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2answers
82 views

How to solve $y'''+2y''-y'-2y= e^x+x^2$?

The equation that needs to be solved is: $$y'''+2y''-y'-2y= e^x+x^2$$ Steps: homogeneous solution and then the particular part.Bbut how do i handle the particular part? Do i need to take them once at ...
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1answer
33 views

First order D.E.

What is the proper solution to $y'=-\sqrt{y}$? Obviously, $2\sqrt{y}=C-x$. But for $x>C$, $\sqrt{y}<0$ and so, $y'<0$. Is the solution then that $y=\frac{1}{4}(C-x)^2$ for $x\le C$ and $y=0$ $...
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3answers
21 views

How to compute the inverse laplace transform of this term? ${-{{3}\over{10}}s-{{1}\over{5}}\over{(s+1)^2+1}}$

So, I have been asked to solve $y'-2y=e^{-t} *cos(t)$ where $y(0)=-2.$ I applied the Laplace transform, getting $$\mathcal{L}(y)={{-2s^2-3s-3}\over{(s-2)((s+1)^2+1)}}$$ I set up a partial fraction ...
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1answer
32 views

Matrix differential equation of the form $X'=CX$

Let $n \in \mathbb{N}^{\ast}$ and $\mathrm{Sym}(n)$ (respectively $\mathrm{Spd}(n)$) denote the linear space (respectively set) of real $n \times n$ symmetric (respectively positive definite) matrices....
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4answers
87 views

I can't derive the integrating factor of this first order ODE from the Dover textbook

I'm a junior mechanical engineering student. I can't derive the integrating factor of this first order ODE $(x^ 2 - y^2 - y) dx - (x^ 2 - y^2 - x) dy = O$ The textbook provides 5 integrating ...
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15 views

Trying to find a differential equation that has multi time scale oscillatory behaviour

I am trying to find two differential equations of dynamical systems (like some non-linear oscillator, perhaps). My requirement is that they both should have the same initial condition (easy to impose)...
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0answers
17 views

A hyperbolic set which is a unique center manifold

Consider the first order ODE \begin{equation} \dot{x} = f(x), \end{equation} with $x \in \mathbb{R}^3$ and $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ a smooth function. We assume that the ODE only has ...
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1answer
284 views

Differential Equations to solve the changing radius of a drop of liquid

This is the question: "Your lab partner leaves a drop of bleach on the lab bench, which takes the shape of a hemisphere. The drop initially has a radius of 1.6mm, and evaporates at a rate ...
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1answer
37 views

Solutions to Sturm-Liouville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Sturm-Liouville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
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1answer
78 views

How to solve $y'' + y = -2\sin(x)$?

I don't know how to find the particular solution of $$y'' + y = -2\sin(x)$$ I started with $$y'' + y = 0$$ to find the homogeneous form $$A\cos(x)+B\sin(x)$$ But now i am stuck.
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Intuition on the necessity of the Lipschitz condition and a physical example of an ODE

The Picard-Lindelöf theorem states that the initial value problem $$ y'(x) = F(x,y(x)), \ y(x_0) = y_0$$ will always have a unique solution on some closed interval containing $x_0$ assuming that the ...
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2answers
52 views

How to tell if you have specified sufficient initial data for a differential equation?

I recently learnt that the following 'wave equation' is not well-posed $$ \begin{cases} \partial_{tt}u=\partial_{xx} u, & (0,1)\times\mathbb R\\ u(t,0)=u(t,1)=0,&t\in [0,1] \end{cases} $$ ...
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17 views

determine the indical equation for the following differential equation

$x^3y''+(\cos2x-1)y'+2xy=0$ One have to make the anstaz that $y= x^m \sum_{j=0}^{\infty}a_j x^j$ and I have been solving problems like this before, but for this one what really confuses me is what to ...
2
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1answer
43 views

Finding a Relation Between Two Sequences

Consider the following recurrence relation for $C_i(r)$s $$\begin{align} &C_0(r)=r-r_2 \\ &q(r+n)C_n(r)+\sum_{k=0}^{n-1}[(r+k)\alpha_{n-k}+\beta_{n-k}]C_k(r)=0, \qquad n\ge1 \end{align} \tag{...
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1answer
401 views

Integral Equation without solution?

working on a physical problem I arrived at the following equation $$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
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41 views

PDE basic traffic flow problem

I am analyzing a basic example of traffic flow presented here http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf and have a question to the last transition in the traffic flow equation ...
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4answers
67 views

Prove using induction the following equation is true.

If $$(1-x^2)\frac{dy}{dx} - xy - 1 = 0$$ Using induction prove the following for any positive integer n$$(1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+3)x\frac{d^{n+1}y}{dx^{n+1}} - (n+1)^2\frac{d^ny}{dx^n} ...
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Relationship between differential equation and difference equation.

$(i)$ Find the solution to $y''-y'-2y=0$ that satisfies $y(0)=1$ and is bounded as $x\rightarrow\infty$. $(ii)$ Solve the related difference equation $(y_{n+1}-2y_n+y_{n-1})-\frac12 h(y_{n+1}-...
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18 views

Evaluate Higher Order Homogenous Differential Equation $ (D^4 - D^3)y=0; y(0) = 1 = y'(0), y''(1) = 3e, y'''(1) = e $

I'm getting following values of constants: $C1 = 0$ $ C2 = 0$ $C3 = e$ $ C4 = 1$ But in my solution manual of the book, the constant values are coming like this: $C1 = 0$ $ C2 = 0$ $C3 = 1$ $ ...
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8 views

Differential form of surface integral equation

Considering a scalar field in a plane (pressure vs. location) $P({\rm r})$ where ${\rm r}=(x,y)$ then the following surface integral gives the surface deformation due to the pressure in the elastic ...
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2answers
41 views

How to solve this using power series method $\left(x^2+2\right)y''\:+\:xy'\:-\:y=0$

$\left(x^2+2\right)y''\:+\:xy'\:-\:y=0$ What's next after this $\sum _{n=2}^{\infty }\:n\left(n-1\right)a_nx^n+2\:\sum _{n=0}^{\infty \:}\left(n+2\right)\left(n-1\right)a_{n+2}x^n+\sum _{n=2}^{\infty ...
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30 views

A Simpler way of Thinking About Integrability

I have recently been looking into the concept of integrability, and the more I look into it, the less sense it seems to make. There seems to be several kinds of integrability and their definitions ...
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2answers
2k views

Modeling a chemical reaction with differential equations

The problem says: Two chemicals $A$ and $B$ are combined to form a chemical $C$. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of $A$ and $B$ ...
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1answer
134 views

Differential equation based on chemical kinetics

In chemical kinetics, the law of mass action gives us reaction rates of the form $$r=k x^a y^b$$ where $r$ is the time derivative of either $x$ or $y$ times a constant $$r=-\frac{dx}{\beta dt}=-\...
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3answers
126 views

Should we re-define Sine?

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle. However I think these ...
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1answer
22 views

Reduction of order of 2nd order ODE (without $y_0$ or $y_1$ provided) [on hold]

Please help to reduce $y''+y'=0$. There are no initial condition nor a solution to start from. From a similar question, I've tried a suggestion to let $W = y'$. But don't come to an agreeable ...
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1answer
42 views

Show that there aren't negative eigenvalues.

I've been trying to solve this Sturm-Liouville theory problem. Show that the problem: $$\left\{\begin{matrix} y''+(x+\lambda)y = 0\\ y(0)=0\\y(1)=0\end{matrix}\right.$$ doesn't have ...
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1answer
19 views

Use the Wronskian to determine a first-order inhomogeneous differential equation for $y_2(x)$.

The function $y(x)$ satisfies the linear equation $y''+p(x)y'+q(x)=0$. The Wronskian $W(x)$ of two independant solutions, $y_1(x)$ and $y_2(x)$ is defined as $\begin{vmatrix}y_1 & y_2\\ y'...
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What is wrong with my formulas for a mathematical model of a double pendulum?

I wanted to create a computer simulation on Matlab, using a model for a pendulum from this study (A double pendulum model of tennis strokes. Rod Cross. Uni of Sydney, 2006) - Link I wanted to use the ...
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1answer
80 views

Quasilinear second order ODE

Consider a smooth $u\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $$ u^{\prime\prime}+a\left(u^{\prime}\right)^{2}+bu=0\text{ on }\mathbb{R} $$ with $$ u^{\prime}\left(x\right),u^{\prime\prime}\...
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2answers
79 views

Chemical reaction modeled by a differential equation

I am badly stuck on the question... so asking some help :) Consider a chemical reaction in which compounds $A$ and $B$ combine to form a third compound $X$. The reaction can be written as $$A ...
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1answer
74 views

Inner Product Examples, what is the points?

Example: For $ -\pi<x<\pi$, $$x =-2 \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx)$$ and $$x^3 =-2 \sum_{n=1}^{\infty} \left( \frac{\pi^2}{n}-\frac{6}{n^3} \right)(-1)^n \sin(nx)$$ by ...
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1answer
26 views

How to compute the slope for a 3 or multi-dimensional equation.

If I have an equation $Z=X^2+Y^2+3X+6Y+5$ and want to find the slope at the point $x=2$, $y=1$. How do we compute it? I know for a two dimensional equation we can compute it by differentiation of $Y$ ...
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3answers
60 views

Finding Explicit Form of Function Defined by Definite Integral

Let $$f(y) = \int_{-\infty}^{\infty} e^{-x^2} \cos (xy) \> dx$$ One can show that $$f'(y) = - \int_{-\infty}^{\infty}xe^{-x^2} \sin (xy) \> dx$$ I'm interested in making an ODE involving $...
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0answers
24 views

Fundamental Set of Frobenius solutions

Consider the equation $$x^2 (\alpha_0 + \alpha_1x + \alpha_2x^2)y'' + x(\beta_0 + \beta_1x + \beta_2x^2)y' + (\gamma_0 + \gamma_1x + \gamma_2x^2)y = 0$$ Define $$p_j (r) = \alpha_jr(r − 1) + \beta_jr ...
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2answers
117 views

Difficult engineering second order DE, any pointers?

I have the following engineering DE: $$rR''+R'+\alpha r(R^2_0-r^2)\lambda^2R=0$$ Where $R(r)$ is Real, $r \geq 0$, $\alpha >0$. Boundary conditions $R(R_0)=0$ and $\Big(\frac{dR}{dr}\Big)_{r=0}=...
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4answers
94 views

Solving Differential Equation -trouble-

Given the equations: $$\dfrac{dy}{dx} - \dfrac{1}{x}y = \dfrac{1}{x^3}y^2,$$ and $y(1) = 1$, I am supposed to solve for $y$. Eventually through my work, I find $$x^{\color{red}1}v = -\int \dfrac{1}{x^...
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0answers
19 views

Solving ODE with irregular singular point

I want to solve the following ODE $$x''(z)+ \frac{\frac{d}{dz} \left(\frac{f(z)}{z^2}\right)}{\frac{f(z)}{z^2}}x'(z)+\frac{\omega^2}{(f(z))^2}x(z)=0$$ where $$f(z) = 1- 4 \left(\frac{z}{z_*}\right)^...
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1answer
49 views

General solution of a nonlinear differential equation

Nonlinear differential equation gone beyond my field of expertise but I'd like to know the details of a problem and to do that I should know the general solution of the following nonlinear ...