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Questions tagged [differential-equations]

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

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Solving Differential Equation for the Dissolution of a Rectangular Prism

So I've developed a differential equation to describe the dissolution of an object in a liquid via a chemical reaction. The equation is as follows $$\frac{dm}{dt}=-cS(t)$$ where m is mass, t is time,...
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Continuity of $\nabla Z$ when the equation for $Z$ has a discontinuity

I am working in a problem of fluid dynamics (diffusion flames, exactly) and I am struggling to prove the continuity of a variable. The function $Z:R^n\to R$ is continuous, but not necessarily smooth. ...
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1answer
13 views

Reduction order

Good afternoon, I am going over reduction of order method in solving a second order linear equation. As show in the example in this link: Following the method, I arrived at $w' - w = 0$ which is a ...
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1answer
15 views

“Unit sample response” relation to “step reponse” $(u[n] \to \delta[n])$

Book says: $$\delta[n]=u[n] - u[n-1]$$ Therefore, the unit sample reponse $h[n]$ is related to unit step reponse, $s[n]$, as follows: $$h[n]=s[n] - s[n-1]$$ My question is, how to prove this ...
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1answer
33 views

Why integral curves cannot be tangent to each other?

I am watching Lecture 1 of the differential equation course on MIT OpenCourseWare. The teacher said, for $y'= f(x,y)$, if $f(x,y)$ is continuous around a point $(x0,y0)$, it would guarantee at least ...
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Gauge transformation of differential equations I

This is a follow-up question to Gauge transformation of differential equations. . Let $y(x)$ be a solution to the following ODE: \begin{eqnarray} y^{''}(x) + a_1(x) y^{'}(x)+a_0(x) y(x)=0 \end{...
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2answers
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Is there an intuitive way to understand $\frac{x\space dy-y\space dx}{x^2+y^2}=d(\arctan\frac yx)$

My book says- $$\frac{x\space dy-y\space dx}{x^2+y^2}=d(\arctan\frac yx)$$ Specifically I am solving differential equations where I may have to transform the LHS into the RHS. Is there an ...
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Hai… my problem is i dont know how to solve this question because I confuse to use any method

$e^{n+y}=1+5\sqrt{y^2}-\ln\left[\frac{y^5(6n^2+1)}{\sqrt{2n^3-4}}\right]$ Find $\frac{dy}{dn}$ when $n=0$ and $y=1$. How can I approach this problem? (original link: https://i.stack.imgur.com/W9Mym....
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3answers
27 views

Solving Riccati ODEs

I have recently started studying the methods behind solving different types of differential equations and have made it to Bernoulli with no problems thus far. However, as I was investigating Riccati ...
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1answer
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Determine p(z) by solving dp/dz = -(1/λ)(p)

Determine p(z) by solving the differential equation dp/dz = -1/λ*p where λ is a constant, and find the particular solution that satisfies the initial condition p(0) = P , where P is a constant. I've ...
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Is the following integro-differential equation solvable? $\frac{dx}{dt} = a_0 + (a_1+a_2x)e^{-a_3 \int_0^t x(s)} + a_4x(t)$

I have an integro-differential equation of the following form: $$ \frac{dx}{dt} = a_0 + (a_1+a_2x(t))e^{-a_3 \int_0^t x(s)} + a_4x(t) $$ In an attempt to solve this equation, I transformed it first ...
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Why the set S(t) is the subspace of the null space of A?

The picture below is on the page 5 of the paper On linear differential-algebraic equations and linearizations. What I have learnt from the pic is as follows: $N(t):=ker A(t)\subset \mathbb{R}^m$ is ...
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On solving a system of equations to find a curve based on its curvature and torsion.

Quoted from Pressley's Differential Geometry : How to show that the three mentioned equations have a unique solution with initial conditions? Since $k=k(s)$ and $t=t(s)$ are functions not ...
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Emden–Fowler equation how to solve?

I am trying to solve the equation $$\frac{y''}{y'}=\frac{\alpha}{r y^2}$$, which is of the form of the Emden–Fowler equation. I found a solution in chapter 2.5.2.5 of Handbook of Exact Solutions for ...
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Falling Object - State-space and “measure” position

Have the following equation: $m\cdot a(t)= m\cdot (-g) - f\cdot v(t)$. Sum of forces on object consists of gravity*mass and a constant $f$ proportional to the velocity of the falling object. How do ...
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1answer
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Solve the equation similar to the Kummer equation

In my calculations, I arrived at the differential equation below. What should I do to solve this equation, which is similar to the Kummer equation? Can the Kummer equation be obtained by changing the ...
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1answer
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Weber-Hermite differential equation

I was solving a quantum mechanics problem (harmonic oscillateur) and i need to solve this Weber-Hermite differential equation in an analytic method: $$y"-x^2(y)=0$$ I know the solution of this ...
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20 views

Lagrange Interpolation Coefficients/ Frobenius Covariant

I am having problems trying to show that $$ \sum_{k=1}^{n} L_k(A)=I$$ where A is a $nxn$ matrix,$I$ is the $nxn$ identity, and $L_k$ is the Lagrange interpolation coefficient defined as: $L_k(A)=\...
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Differential Equation linear, separable, neither or both?

I am stuck on deciding whether or not the differential equation $$\frac{dy}{dx} = \frac{1}{x} + \frac{1}{y}$$ is linear or separable? I believe it is linear although I am confused on the value of $P(...
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Find a particular solution for this linear system

$\mathbf{x'}=\begin{bmatrix}3 & 1 \\ 0 & 3\end{bmatrix}\mathbf{x}+\begin{bmatrix} e^{3t}\\e^{3t}\end{bmatrix}$ My attempt: We first find a solution to the associated homogeneous system $\...
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1answer
37 views

Solve ODE for real free falling: $y(x)^2\cdot y^{\prime\prime}(x)]=4\cdot 10^{14}$

I am trying to describe the position of a free falling ball by gravity: if $x$ is the time in seconds, $y$ is the position of the falling ball, $y^{\prime\prime}(x)$ is its acceleration then $$ F=G\...
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Separation of variables for nonhomogeneous PDE

I need to solve the equation below using separation of variables. $$\frac{\partial f(x,y)}{\partial x} - \frac{\partial f(x,y)}{\partial y} = 2$$ The thing is, i've always done with $0$ after the ...
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2answers
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Finding the roots of a characteristic polynomial

Main aim is to find the lowest order equation with the solution: $$y(x)= 2 \cosh(x) + 3 e^{-2x} \sin(x)$$ Now, I am trying to find the roots to form the characteristic polynomial from which I get the ...
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Can someone find this inverse (and solve Burger's equation)?

My question regards the following function: $f(x) = \begin{cases}\frac{x}{\log\left(\frac{x+1}{x-1}\right)}&\text{ for }x>1,\\ 0&\text{ for x=1}\end{cases}$ which arose in circumstances I ...
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2answers
36 views

Solve: $xu_x+(x+y)u_y=1$

Solve the following PDE $$xu_x+(x+y)u_y=1$$ when $$u(1,y)=y$$ using method of characteristics and find the projections of the characteristics on the xy plane $$\begin{cases} a=x\\ b=x+y\\ c=1 \...
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1answer
42 views

Equation of the Surface of Fusée Barrel

The fusée is a device used in certain mainspring-driven clocks, to level-out the torque that the mainspring applies to the movement. A mainspring without correction produces a torque that declines ...
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Nonlinear Operator compact but not strongly continuous

can someone help me on this problem? Consider $V=L^2(0,1)$ and $g \in V \setminus ${0}$ $ and let $A: V \rightarrow V^\ast$, $v\mapsto Av := g ||v||²$ be an nonlinear Operator. Prove that A is ...
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Show that $p$ is a stationary solution

Let $\Omega\subseteq\mathbb R^n$ be open and $f:\Omega\to\mathbb R^n$ a continuous vector field. Show that if for some $p\in\Omega$ there is a solution $\gamma:]a,\infty[\to\mathbb \Omega$ of $x'=f(x)$...
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1answer
32 views

Eigenvalue of a given operator

If $u_0$ is a positive solution of $$ -\frac{1}{2}\frac{d^2u}{dx^2}+\lambda u -u^3=0 $$ Then how to show $-3\lambda$ is a eigenvalue of $$ Lu=-\frac{1}{2}\frac{d^2u}{dx^2}+\lambda u -3u_0^2 u $$ and ...
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Find the general solution to the ODE $x\frac{dy}{dx}=y-\frac{1}{y}$

I have been working through an ODE finding the general solution and following the modulus through the equation has left me with four general solutions, as shown below. Online ODE solvers, however, ...
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1answer
59 views

Power series solution to a differential equation

If $f(\rho)$ satisfies $\frac{df}{d\rho}=\frac{f(2\rho)}{2f(\rho)}$, I am trying to derive the form of $f(\rho)$ by using a power series expansion $f(\rho)=\sum a_n \rho^n$ and show that $f(\rho)$ can ...
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1answer
67 views

Using Parseval's identity to show that $\frac{\pi^2}{8}=1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+…$

By considering the Fourier sine series on the interval $[0,\pi]$ for $f(x)=1$, show that $$\frac{\pi^2}{8}=1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...$$ I am having trouble computing the Fourier ...
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1answer
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Confusion with how to handle constant in differential equation

I'm trying to learn to solve differential equations. Currently in class we are discussing integration factors. Here is one of the problems on the homework - $$(3xe^y+2y)dx + (x^2e^y+x)dy=0\tag{1}$$ ...
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Finding eigenfunctions and eigenvalues from a differential equation

Consider the differential equation $$X''(x)+\lambda X=0$$ on $0 \leq x \leq 1$with boundary conditions $$X'(0)+X(0)=0 \ \ \ \ \text{and} \ \ \ \ X(1)=0.$$ I have a few problems here that I think ...
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special partition function

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$. $$ \mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...
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1answer
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determine the stability of an equilibrium point(x,0).

I would like to determine the stability of equilibrium point $(x,0)$ of the differential equation $\dot x = Ax$ $ A= \bigg[ \begin{matrix} 0&0\\0&a \end{matrix} \bigg] $ and $ a >0 $ ...
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1answer
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proof (Legendre polynomial )

I'm stuck, couldn't figure it out. I appreciate your help. Show that : $$\frac{1}{2} \ln \left |\frac{1+x}{1-x} \right |=\sum_{n \text { odd}} \frac{(2n+1)}{n(n+1)}\mathcal {P_n(x)}$$
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Conclude $u$ cannot obtain its maximum if it fulfills a linear differential inequality $Lu \geq 0$

I have a function $u \in C^2((a,b)) \cap C^0([a,b])$ and a bounded function $g : (a,b) \rightarrow \mathbb{R}$, whereas $(a,b) \subset \mathbb{R}$ Now consider the linear operator $L := \frac{d^2}{...
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Show that $f''(x)=e^xf(x)$ with $f(a)=f(b)=0$ makes $f\equiv 0$ $\forall x \in [a,b]$

Define $f \in C^{2}\left[a,b\right]$ satisfying $f''(x)=e^xf(x)$. Show that $f''(x)=e^xf(x)$ with $f(a)=f(b)=0$ makes $f\equiv 0$ $\forall x\in [a,b]$. Actually, I figure out a solution as follows: (...
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1answer
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In homogeneous ODE

If the solution to an ode is an exponential of x multiplied by $\sin x$ , what “guess” do i use for $y_p$ in order to solve the equation? I have tried using the guess $e^x(A\sin x + B\cos x)$ but i ...
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Setting up a predator-prey model with diffusion

I'm trying to set up a predator-prey model, where the predator affects the distribution of the prey. Effectively, over time the prey has moved from areas of high concentration of the predator to areas ...
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phase portrait of nonlinear systems without nonlinear method

\begin{align}dx/dt&= y\\ dy/dt&= x + 2x^3: \tag{a} \end{align} (b) Find the orbits of the given nonlinear problem (a) and draw its phase portrait. (c) Show that there are exactly two orbits ...
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Find a solution bounded near $x=0$ of the following ODE

Fine a solution bounded near $x=0$ of the following ODE $$x^2y''+xy'+( \lambda ^2x^2-1)y=0$$ my attempt : this is Bessel's equation so let $u=\lambda x$ then $y(x)=y(\frac{u}{\lambda})$ Also $...
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Solving a system of partial differential equation equivalent to finding integral curves?

Consider the following system of partial differential equations defined on some neighborhood $U$ of the origin in $\mathbb{R}^2:$ $$z_x = F(x,y,z(x,y)), z_y = G(x,y,z(x,y))$$ where $F$ and $G$ are ...
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Bernoulli equation with $e^y$ term

I'm having a hard time solving the following differential equation: $$(x^3 + e^y)y' = 3x^2$$ I'm familiar with the approach of introducing $z=y^{1-2}=y^{-1}$, but that doesn't do the trick. Am I ...
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1answer
20 views

Discrepancy in solutions of differential equation?

The differential equation at hand is this : $$ \frac{\text{d}\psi}{\text{d}x}+2\tanh(x)\,\psi\left(x\right)=0\ $$ And what I have tried is this : $$ \int_{}^{} \frac{\text{d}\psi}{\psi}=-2\int_{}^{}...
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Using Picard-Lindelof to find a solution to $y'(t,y(t))=t+\sin(y(t))$ where $y(2)=1.$

Consider the initial value problem $y'(t,y(t))=t+sin(y(t))$ with initial condition $y(2)=1$. Find the largest interval $\mathcal{I}\subset \mathbb{R}$ containing $t_0=2$ such that the problem has a ...
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2answers
95 views

Integral of $\int_0^{\infty} \frac{\sin^2(x)}{x^2+1}dx$ using Feynman integration.

Using $$I(t) = \int_0^\infty \frac{\sin^2(tx)}{x^2+1}dx$$ I want to know how to get an answer using Feynman integration and the Laplace transform of a differential equation. The correct answer is $\...
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1answer
46 views

How to solve an ODE in Sturm-Liouville form

I am attempting to solve the following ODE: $\frac{d}{dz} \bigg[F'(z) \bigg(\frac{z-1}{z}\bigg)^2\bigg] = \frac{2(z-1)F(z)}{z^4}$ with the conditions that both $F(z)$ and $F'(z)$ approach 0 as $z$ ...
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1answer
46 views

Existence of periodic orbit of a nonautonomous system

Consider the system $$\left\{ \begin{array} & x' & = & y\\ y' & = & -2y-x^3+1/\sqrt{27} + \epsilon \cos t\\ \end{array}\right.$$ Prove that, if $0<\epsilon \ll 1$, there ...