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

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Exercise at Differential Equations: boundary value problems

$ Let\quad U\quad be \quad a \quad smooth \quad solution \quad of \quad the \quad following\quad boundary \quad value\quad problem: $ $ -cU'+ (F(U))'=εU''\qquad U(-\infty)= A \quad and \quad U(+\...
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integrate equation

I am trying to integrate this equation, however I am not sure which method would be best. $\frac{\dot{a}}{a} = -2 \alpha \frac{\dot{M_1}}{M_1 + M_2}$ All the variables $a, M_1, M_2$ are time ...
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Explaination of this particular approach to form differential equation.

Consider we need to form differential equation of the following: $Ax^2 + By^2 = 1$ where A & B are constants. Now one approach I know of is two differentiate this equation 2 times (since 2 ...
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2answers
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Nonuniqueness of Stochastic Differential Equation

Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and $$dX_t = \mu(t,X(t)...
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1answer
332 views

A differential equation (nonlinear First-Order)

how to solve this equation: $(Px-y)(Py+x)=h^2P$ that $P=\frac{dy}{dx}$ and $h$ is a constant.
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How to solve complex-valued , inhomogeneous second order differential equations?

I am trying to find a general method to solve the following complex-valued , inhomogeneous second order differential equation $$ -a(x) u''(x)+b(x)u'(x)+c(x)u(x)+d(x)\bar{u}(x)=f(x),\quad x\in\mathbb{...
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How to start an eigenvalue problem

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems... please give me a hint or a clue ,cause I ...
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1answer
695 views

diffusion equation plot (matlab or maple)

The advection diffusion equation is the partial differential equation $$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} - v \frac{\partial C}{\partial x}$$ with the boundary ...
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24 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
27 views

Differential equation function. Cant crack this one up. Tomorrow test

The equation is. $\cos(x) \cdot y'+\sin(x) \cdot y = 2(\cos(x))^3 \cdot \sin(x)-1$ a) Find all the solutions of the associate homogeneous equation. Let $S_h$(homogeneous) denote such a set of ...
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1answer
18 views

Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
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Airy's Equation For Series Solutions to Second Order Linear Equations

I am trying to do an example in the textbook for series solutions to second order linear equations, and I am not quite understanding the recurrence relation as follows: I understand and follow the ...
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1answer
58 views

Solutions to the differential equation $x(x+1)yy' = xy + 1$

I am having trouble solving the linear equation $x(x+1)yy' - xy - 1 = 0$ I will list the steps I followed: (I'm sure I have made some huge mistake.) Divide by $x(x+1)$ $yy' - y/(x+1) - 1/x(...
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Separation of variables and h(y) = 0

Consider the following differential equation $$ y' = g(t) h(y) \tag{1} $$ Solving this by separation of variables, we consider the following cases $ h(y) \neq 0\ \forall\ t$, so we have $ \frac{1}...
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How to find a suitable function for Dulac's criteria in this example?

I have a system of odes $\dot{\mathbf{x}} = \mathbf{f(x)}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $\mathbf{f(x)}$ is defined below: $$\dot{x} = x- y - x^{3}, \qquad \dot{y} = x+y-y^{3}$$ I would ...
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Direction of a curve given by ODE

Let $a,b \in C(\mathbb{R^2})$ be bounded and $(x_0,y_0) \in \partial B_1(0)$. Consider the ODE system $$ \begin{cases} x'(t)=a(x,y) \\ y'(t)=b(x,y) \\ x(0)=x_0 \quad y(0)=y_0 \end{cases} $$ We know ...
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Solving$ x(x-1)\ddot y-x\dot y+y=x(x-1)^2$ by using integrating factor

Solving $x(x-1)\ddot y-x\dot y+y=x(x-1)^2$ given the solution $y_1=x$ I set $y=vx$, have plugged and rearranged. I set my integrating factor to: $$e^\left({\int\frac{x-2}{x(x-1)}dx}\right)$$ ...
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50 views

Finding a parametrization of the solutions of $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$, $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$

I am trying desperately to find a parametrization for the following: $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$ $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$ I tried to devide the equation ...
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3answers
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Solving $(x^2-1)\ddot y-2x\dot y +2y=1$

Solving $$(x^2-1)\ddot y-2x\dot y +2y=1$$ I've solved the homogenous equation: $$y=A(x^2+1)+Bx$$ Where A and B are constants of integration, but I can't for the life of me seem to remember how to ...
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Equation of silhouette from an arbitrary viewpoint

A two parameter $(u,v)$ surface in $\mathbb R^3$ when viewed from a point at infinite distance casts a shadow on any given plane. What ODE/PDE describes its envelope of its silhouetted projection? ...
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1answer
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“Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...
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1answer
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What is the missing step to solve the two ODE?

I was reading an example of an exercise of variational calculus where they get the system of equations: \begin{equation} F_{y'y'}y''+F_{y'z'}z''=0;\;\;F_{y'z'}y''+F_{z'z'}z''=0, \end{equation} where $...
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2answers
41 views

Solved ODE by two different methods; unable to prove the solutions are the same.

I have a first order ODE: $$\frac{dy}{dx}+\frac{2y}{x}=\frac{e^x}{x^2}\tag{1}$$ Noting that this is in the form $$\frac{dy}{dx}+P(x)y=Q(x)\tag{2}$$ So an integrating factor method can be used. ...
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Stability of ODE with periodic coefficients / Periodic solutions

Given the ODE system $$ \left\{ \begin{array}{l} \dot x = -2x - z \cos t, \\ \dot y = x \sin t - y, \\ \dot z = -4z + \sin^2 t. \end{array} \right. $$ I am asked to: 1) Examine the stability of ...
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Difficulties understand the series solution of $(1-x^4)y''-8x^3-12x^2y=0$

Solve: $(1-x^4)y''-8x^3-12x^2y=0$ using the solution: $$y=\sum_{n=0}^{\infty}a_nx^n$$ Let's differentiate y: $$y'=\sum_{n=1}^{\infty}(n)a_nx^{n-1}$$ $$y''=\sum_{n=2}^{\infty}(n)(n-1)a_nx^{n-2}$$ ...
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1answer
48 views

does a linear differential equation have a well defined initial value problem if a term diverges at initial 'time'?

Suppose I have a differential equation like the following: $$\frac{d^2x}{dt^2}+t^2x=0$$ And I've to put initial conditions at $t=-\infty$. Now the $t^2$ bit seems to diverge at $t=-\infty$. Is that ...
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1answer
31 views

Why does a trajectory take infinite time to reach a critical point?

I have to prove that: For an ODE system $x'=F(x,y),y'=G(x,y)$ where $F,G$ are smooth, any trajectory, which doesn't start at a critical point, cannot reach a critical point in finite time. I have ...
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1answer
49 views

Clairaut's form of $(x\frac{dy}{dx}-y)(y\frac{dy}{dx}+x)=a^2\frac{dy}{dx}$

Question is to find the Clairaut's form of differential equation $$(x\frac{dy}{dx}-y)(y\frac{dy}{dx}+x)=a^2\frac{dy}{dx}$$ I know clairaut's equation is of the form $y=x\frac{dy}{dx}+f(\frac{dy}{dx})$...
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(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
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1answer
289 views

Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
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1answer
33 views

Determine the solution of the following system of Differential Equations

Determine the solution of the following system: $$\dot{x_{1}}=-x_{1}$$ $$\dot{x_{2}}=-x_{2}+x_{1}^{2}$$ $$\dot{x_{3}}=x_{3}+x_{1}^{2}$$ The first equation clearly has solution $x_{1}(t)=c_{1}e^{-t}$...
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1answer
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Can someone explain where this comes from for the cauchy-euler equation in the case of double roots

I am not sure how the following was derived in my textbook given by I understand that the solution we are seeking is $y = x^r$ and the fact that we obtained the first solution as $y = c_1 x^{r_1}$ ...
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1answer
956 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
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3answers
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Differential Equation Initial Value Problem

Here is a pretty standard initial value problem that I'm having a little trouble with. $$(\ln(y))^2\frac{\mathrm{d}y}{\mathrm{d}x}=x^2y$$ Given $y(1)=e^2$, find the constant $C$. So I separated and ...
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2answers
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Differential equation $\left(x^2+xy\right)y'=x\sqrt{x^2-y^2}+xy+y^2$

I am not sure which type of differential equation this falls into: $$\left(x^2+xy\right)y'=x\sqrt{x^2-y^2}+xy+y^2$$ any hints? P.S. I first tried reornazing it so I have $y'$ alone, and hoping that I ...
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System of N-1 first order ODEs

I've to elaborate this system of N-1 first order ODEs for i = 1,2,...,N and n =1,2,...,N. $ (z_3)_n^.(t)= \sum_{i=1}^{N-1} (C_3)_{ni}(z_3)_i(t)+ (z_1)_n(t)\sum_{i=1}^{N-1} (A_3)_{ni}(z_3)_i(t)$ For N=...
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Need someone to refer through via e-mail for math help? [on hold]

Alright, so I am a third-year mathematics student, and I took a differential equations course last semester, and failed. I found the lectures to be very vague and too problem-specific, so I would go ...
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Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$

In order for the question that I have to make any sense I must first include some background information as given in my textbook: The standard form of Bessel's differential equation is $$x^2y^{\...
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Verify $y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$ is a solution to $y^{\prime\prime}+9xy=0$

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...
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28 views

Model for headphone quality as a function of price

I've looked for any mathematical model that shows headphone quality as a function of the price of the headphones, but have not found anything yet. Of course, "quality" is quite subjective, which could ...
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1answer
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Difficulty understanding step in Kac's proof of Feynman-Kac Theorem

I am trying to understand a proof of the Feynman-Kac Theorem, as set out in Mark Kac's 1949 paper 'On Distributions of Certain Wiener Functionals'. Kac defines a series of independent and ...
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How to solve the differential equation: $x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18x$

How to solve the differential equation $$x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18x$$ I think I could let $u=xy$ but I don't know how to proceed it.
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System of nonlinear differential equations [on hold]

$$\frac{d^2x}{dt^2}=\frac{Ax}{{(x^2+y^2)}^{3/2}},$$ $$\frac{d^2y}{dt^2}=\frac{Ay}{{(x^2+y^2)}^{3/2}},$$ $$A=const,$$ $$\frac{dx}{dt}_{t=0}=0, \frac{dy}{dt}_{t=0}=0,$$ $$x_{t=0}=2d, y_{t=0}=d.$$
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How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
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Differential Equations (Coffee)

This is a long post so bear with me until I get to the part where I am stuck on! :) Question: The author of a popular detective novel drinks black coffee to help him stay awake while writing. ...
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2answers
65 views

Can you find this tough Integrating Factor?

$$ u\left(du − dv\right) + v\left(du + dv\right) = 0 $$ Can you get this equation in a form by which the integrating factor can be found using the standard algorithm? Expression $\;\displaystyle \mu ...
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2answers
40 views

Could someone please verify whether or not this is a book error?

Below is a short extract for which I believe there may be an error: I think that equations $(22.3)$ and $(22.4)$ have been written out wrongly, they should be ...
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27 views

Solving a system of N-1 Ist order ODEs by Euler's Method

In order to solve a system of N-1 first order ODEs by Euler's Method For N = 4; t=0, h= 0.1, x= 0.1 should the Euler formula be? $U_n(t+h) = U_n(t) + h F_n(x_n, t_n)$ for n = 1, 2,..,N-1 but we ...
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0answers
44 views

$h$-principle for Legendrian immersions

It is "folklore" that continuous curves in contact 3 manifolds can be approximated by Legendrian curves and it seems that this follows from Gromov's $h$-principle for Legendrian immersions (in ...