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

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Solution of Homegeneous Differential Equation

"What if the M and N are of with the different degree. Is the solution exist?" If $M(x,y)dx+N(x,y)dy=0$ are homogeneous function and of the same degree. What if the M and N are of with different ...
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3answers
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Find the value of $y(1)$ of the ODE $y'+y=|x|$.

Let $y$ be the solution of $$y'+y=|x|$$ for $x\in\mathbb{R}$ and $y(-1)=0$. Then the value of $y(1)$ is $\frac{2}{e}-\frac{2}{e^2}$ $\frac{2}{e}-2e^2$ $2-\frac{2}{e}$ $2-2e$ I don't know what to ...
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1answer
54 views

Bessel Function of the first kind

Could you please help me understand how to prove $$J_{(1/2)} (x) = \sqrt{\frac2{\pi x}}\cdot \sin⁡ x$$ using, $$J_p (x) = \sum_{(n=0)}^\infty \frac{(-1)^n}{(n! \Gamma(n+p+1) )} \left( \frac x 2 \...
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1answer
60 views

Can't get rid of integrals solving this differential equation.

Assuming $\mathbf{A}\equiv \vec A$ , $\dot q\equiv \frac{d}{dt}q$ ,and $\ddot q\equiv \frac{d^2}{dt^2}q$ , And Using the Right-hand Cartesian coordinate system with base vectors $\mathbf{\hat i\, \...
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5answers
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Differential Equation [closed]

A new car is purchased for $35000$ dollars. During the next $10$ years, the rate of change in the value of the car is given by the differential equation $dx/dt = -kx$, where $x$ is the value of the ...
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1answer
47 views

Calculating the particular solution of the following nonhomogeneous system

The name of the game is systems of differential equations and matrix exponentials. I have the following problem: Apply the formula $\vec x(t) = e^{At} \vec c + e^{At} \int e^{At} \vec f(t) dt$ (where $...
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0answers
43 views

Solutions of Homogeneous Differential Equations

an equation said to be homogeneous if they are on the same degree, "What if the functions are in the different degree? Is it homogeneous or the solution exist ? ...
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1answer
36 views

Is there a method to quantify the return time to equilibrium?

I'd like to know how fast (or slow) a dynamical system will return to equilibrium, especially if it is near an unstable equilibrium. The question is vague, but that is intentional. I'm looking for ...
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1answer
46 views

Laplace Transform for Solving Differential Equation

I solved the following task, but since I am new in this field I need to check if it is correct or if there is anything I am missing or doing wrong. Task : Solve differential equation using Laplace ...
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1answer
33 views

if some set of functions are not linear, how can their composition be a linear combination that solves a linear second order ODE?

What I am trying to arrive at is, If a function is the composition of some set of non-linear functions, how can that composition be linear? Here is the example I has been working through. Example 1: $...
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2answers
24 views

Need help with substitution

I need help with substituting to differential equation: $$y\,y'_x=\left(ax+b\right)^{-2}\,y+1$$ The substitution $a\xi=-\left(ax+b\right)^{-1}$ leads to an equation of the form $y\,y'_\xi=y+\...
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1answer
96 views

Solve a nonlinear system of coupled differential equations

I have this system of differential equations which describes the motion of a missile launcher model with 5 degrees of freedom: (1)$$(m_w +m_v +m_p)\ddot{y}_w - (m_v + m_p)h_v\ddot{\vartheta}_w \sin(\...
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1answer
104 views

Intuitive explanation of a stochastic PDE

Lindgren et al 2011 connects Gaussian Markov Random Fields (which have fast calculation properties due to the Markov attribute) and Gaussian Processes (which can model many types of data). The ...
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1answer
25 views

Modeling - First Order vs Higher Order Differential Equations

Throughout my engineering education, I've only witnessed modeling using First Order differential equations. Almost all of my Higher Order knowledge was given either via a spring or heat equation or ...
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1answer
48 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 ...
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1answer
68 views

How can I solve $y''=\frac{a}{y^2}$ where a is a (positive) constant?

Actually, I found out a way to solve that, but I can't get rid of complex numbers. And it does not make sense when it comes to complex numbers as the original question that involves this differential ...
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0answers
25 views

Confused about this equation with limits and differentials

I am dumbfounded as to how to solve this particular equation: Given $y(x+h)=y(x)-h(\alpha y(x)+\beta), y(0)=n$, where $\alpha$, $\beta$ and $n$ are constants, find $\lim_{h\to0}y(x)$ I tried solving ...
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1answer
32 views

Non homogeneous Second order ODE

How to solve differential equations of the form: $$\ddot{r}+A(t)r=B(t)$$ In particular I would like to solve the following differential equation: $$\ddot{r}-\omega^2r=\frac{V}{t}$$ Where $\omega$ and ...
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2answers
48 views

Help in solving the differential equation $y'(x)=y(x) + \int_0^1ydx$

If $y(x)$ satisfies the eqn $$y'(x)=y(x) + \int_0^1ydx$$ Where $$y(0)=1$$ Then what will be the value of $$y(ln\frac{11-3e}{2})$$ I tried differentiataing both sides wrt $x$, I got $$y''(x)= y'(x)$$ ...
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3answers
45 views

Solving differential equation of third degree

If differential equation of the curves $$c(y+c)^2 = x^3$$ where 'c' is an arbitary constant is $$12y(y')^2 + ax = bx(y')^3$$ What is the value of a+b? I tried differentiating the curve given and ...
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0answers
34 views

Request for Introduction to Complex valued PDE's and their applications in math and physics

On Mathematica.stackexchange there is a recent question concerning PDE's with complex damping coefficients, see http://mathematica.stackexchange.com/questions/119870/can-i-use-operators-of-the-form-...
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0answers
32 views

Under what condition do the limits of the functions defined below exist?

I have a system of autonomous differential equations: $$\mathbf{\dot{x}}={{\mathbf{S}}^{T}}\mathbf{f}\left( \mathbf{x} \right)$$ where $\mathbf{x}:\left[ 0,\infty \right)\to {{\left[ 0,\infty \...
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0answers
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Non-uniqueness of solutions to the steady-heat equation on the disk that do not converge uniformly to the boundary

According to exercise 18, chp. 2, of Stein & Shakarchi's Fourier analysis, $\frac{\partial P_r(\theta)}{\partial \theta}$ is a solution to the steady-heat equation that converges only pointwise to ...
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1answer
31 views

Reduction of order in ODE

$$x^2 y'' - 3 x y' + 4y = 0$$ where $y_1=x^2 \ln (x)$ is a solution. I need to use the given solution $y_1$ to find a second order linearly independent solution. So $y = y_1 v$ and $y =v x^2 \ln (x)$...
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0answers
16 views

How do I plot this second order distribution in matlab?

http://imgur.com/ChrocF9 I'm trying to do a problem for heat transfer. I understand what's going on conceptually, but it wants me to do a 2D temperature plot in Matlab and I am stuck. The symmetry ...
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0answers
36 views

Isn't a homogeneous differential equation always solvable trivially?

Consider a homogeneous differential equation (linear or non-linear, of any order), defined in the sense here http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html (the first ...
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2answers
48 views

Definition of a derivative of differential form

While reading a paper I encountered the following: Let $(\mathbf{q,p}) \in \mathbb{R}^{2n}$ be canonical coordinates and let $H: \mathbb{R}^{2n} \to \mathbb{R}$ be a smooth function. The ...
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1answer
57 views

Nonlinear differential equation $(y')^2+xy'-y=0$

What will be the solution of $$(\frac{dy}{dx})^2 + x\frac{dy}{dx} - y = 0 $$ I tried writing $$z=\frac{dy}{dx}$$ Then $$ z^2 +xz - y=0 $$ Differentiating wrt x $$ 2z\frac{dz}{dx} +z +x\frac{dz}{...
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1answer
69 views

Reduction of order differential equations

I need to use the given solution y1 to find a second linearly independent solution (X^2 -1)y" - 2xy' +2y = 0, y1 = x So y = y1v => y = xv y = xv y' = v + xv' y" = 2v' + xv" From here I need to ...
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1answer
74 views

Modeling particles moving through a chamber

Consider the following phenomenon. Particles each traveling with speed $v_i$ metres per second enter a chamber at a rate of $r$ particles per second. Upon entering the chamber, each particle begins ...
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1answer
33 views

Legendre's equation of order n in differential equations

The equation $$(1-x^2)y'' - 2xy'+n(n+1)y = 0$$ is called Legendre's equation of order $n$. I need to show that this equation of order $1$ has $y=x$ as one solution and then I need to use this to ...
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2answers
48 views

second order linear homogeneous equations problem

I need to find a second order linear homogeneous equation with constant coefficients that has the given function as a solution question a) $xe^{-3x}$ question b) $e^{3x} \sin x$ We have learned ...
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1answer
25 views

initial value problem, differential equations

A tank initially contains $50$ gal. of brine, with $30$ lbs of salt in solution. Water runs into the tank at $6$ gal./min. and the well-stirred solution runs out at $5$ gal./min. How long will it be ...
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1answer
25 views

Orbits of solution lie in level sets

Let $F:X\rightarrow\mathbb{R}$ be a continuously differentiable function on an open set $X\subset\mathbb{R}^2$. I want to find an autonomous differential equation $y'=f(y)$ on $X$ such that the orbits ...
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1answer
85 views

Given the $n$th derivative of $f$ at $0$ find $f$.

I want to find: $$\frac{1}{2} \sum_{n=0}^{\infty} \frac{(1/6)^n(2n+2)!}{n!(n+1)!}$$ Where the sum looks like a Taylor series of $f(x)$ at $0$: $$\sum_{n=0}^{\infty} \frac{x^n(2n+2)!}{n!(n+1)!}$$ ...
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0answers
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Invariant measure of Poincare map

Let $M$ be a smooth manifold and let $v$ be a tangent vector field on $M$. Consider a system of ordinary differential equations $$ \dot x = v(x), $$ in local coordinates $x = (x_1, \ldots, x_n)$. ...
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2answers
68 views

How to solve the differential equation $x \dfrac {dy}{dx} = \dfrac {y^{2}}{1 - y\log x}$?

How to solve the differential equation $x \dfrac {dy}{dx} = \dfrac {y^{2}}{1 - y\log x}$ ? Does not seem like an easy one or reducible to LDE. Any suggestions?
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1answer
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Finding the General Solution to a Homogeneous Linear Differential Equation (of second order) with repeated roots. [Proof]

Consider the differential equation of the form $\ddot{y}+A\dot{y}+By=0$, where $A$ and $B$ are constants. Suppose, $A^2=4B$, implying that the characteristic equation of this ODE has repeated roots. ...
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3answers
54 views

Find m. $y=e^{mx},m\in\mathbb R,\frac{d^2y}{dx^2}-3\frac{dy}{dx}-4y=0$

If: $$y=e^{mx},m\in\mathbb R$$ Find m if: $$\frac{d^2y}{dx^2}-3\frac{dy}{dx}-4y=0$$ Differentiating and substituting gives: $$m^2e^{mx}-3me^{mx}-4e^{mx}=0$$ Dividing across by $e^{mx}$ and solving ...
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1answer
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can you explain how it get this? this is from fourier series

I'm studying fourier series and I stumble in this equation. and I confuse can tell or explanation how to get that answer: $a_n= \frac{1}{2} \int_0^2 x \cos(\frac{n\pi x}{2})dx =\frac{1}{2}(\frac{2}{n ...
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1answer
70 views

first order and higher degree differential equation

Can someone please solve this differential equation with detailed solution $$y= 2x\frac{dy}{dx} + y^3\bigg(\frac{dy}{dx}\bigg)^3$$ Thanks in advance
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1answer
48 views

Closed Form Solutions of the Second Order Linear ODEs with Non-Constant Coefficients

I am studying about the linear odes with non-constant coefficients. I know the first order linear ode with non-constant coefficient $$y^{'}(x)+f(x)y(x)=0 \tag{1}$$ has a general solution of the ...
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2answers
76 views

Solve the ODE $yy''=y'$

Solve the ODE $yy''=y'$ Can anyone check my solution? And what is the answer? Thanks. Attempt: \begin{align*} y''=\frac{y'}{y} &\implies \frac{dy}{dx}=\ln|y|+c_1 \qquad\text{by integrating both ...
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1answer
30 views

How to prove Linear Independence of piecewise functions?

Suppose we have two functions: $f_1(x)=x^2,x\geq 0$ and $f_1(x)=0,x\leq 0$ and $f_2(x)=0,x\geq 0$ and $f_2(x)=x^2, x\leq0$. Show that these two functions $f_1(x)$ and $f_2(x)$ are linearly independent....
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1answer
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Differential equation with multiple solutions

The below was a question on exam and I solved it like the below.. $$y^{'}=3t^{2}+3t^{2}y$$ $$\frac{dy}{dt}=3t^2(1+y)$$ $$\frac{dy}{1+y}=3t^{2}dt$$ $$\int\frac{dy}{1+y}=\int3t^{2}dt$$ $$\ln|1+y|= ...
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1answer
24 views

Proof for general solution of homogeneous second-order linear ODE?

In a differential equations class the professor stated that the general solution of a homogeneous second-order linear ODE would be in the form: $$y = c_1y_1 + c_2y_2$$ Where $y_1$ and $y_2$ were ...
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1answer
41 views

Solving telegrapher's partial differential equation

Using the method of separation of variables and writing $u(x,t)=M(x)N(t)$, we can solve the equation $$u_{tt}-\gamma^2 u_{xx} + 2\alpha u_t=0$$ $$0<x<l$$ $$t\ge0, \alpha>0 \text{ (}{\alpha} \...
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0answers
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The zero point $(0,0)$ of $x'=x+y,y'=xy-bx-y$ is stable when $b>1$

Proof when $b>1$, the zero point $(0,0)$ of ODE $\left\{\begin{align}&x'=x+y\\&y'=xy-bx-y\end{align}\right.$ is stable. I couldn't find a proper Lyapunov V function. Let $v=x'$ we has 1st ...
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1answer
32 views

Long time asymptotic of Fokker–Planck equation $\; \partial_tu-\nabla\!\!\cdot\!\left(\nabla u+xu\right) = 0$

Is it true that given a solution to the Fokker–Planck equation $$\partial_tu-\nabla\!\!\cdot\!\left(\nabla u+x\hspace{0.2ex}u\right) = 0,$$ then we have $$ \left\|\frac{u-\rho}{\rho}\right\|_{\...
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0answers
30 views

Euler's method (ODE) code implementation

I am trying to write a program in Python to solve a simple initial value problem with Euler's Method for ODEs. Programming is not my forte at all, so I am having trouble with implementation. I would ...