Questions tagged [differential-equations]

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

27,999 questions
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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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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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“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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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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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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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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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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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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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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$\mathbf{x'}=\begin{bmatrix}3 & 1 \\ 0 & 3\end{bmatrix}\mathbf{x}+\begin{bmatrix} e^{3t}\\e^{3t}\end{bmatrix}$ My attempt: We first find a solution to the associated homogeneous system $\... 1answer 37 views Solve ODE for real free falling:$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 $$F=G\... 2answers 22 views Separation of variables for nonhomogeneous PDE I need to solve the equation below using separation of variables.$$\frac{\partial f(x,y)}{\partial x} - \frac{\partial f(x,y)}{\partial y} = 2$$The thing is, i've always done with 0 after the ... 2answers 45 views Finding the roots of a characteristic polynomial Main aim is to find the lowest order equation with the solution:$$y(x)= 2 \cosh(x) + 3 e^{-2x} \sin(x)$$Now, I am trying to find the roots to form the characteristic polynomial from which I get the ... 0answers 34 views Can someone find this inverse (and solve Burger's equation)? My question regards the following function: f(x) = \begin{cases}\frac{x}{\log\left(\frac{x+1}{x-1}\right)}&\text{ for }x>1,\\ 0&\text{ for x=1}\end{cases} which arose in circumstances I ... 2answers 36 views Solve: xu_x+(x+y)u_y=1 Solve the following PDE$$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 \... 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 ... 0answers 25 views 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$be an nonlinear Operator. Prove that A is ... 0answers 16 views Show that$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)$... 1answer 32 views Eigenvalue of a given operator If$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 ... 3answers 48 views 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, ... 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 ... 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 ... 1answer 23 views 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 - $$(3xe^y+2y)dx + (x^2e^y+x)dy=0\tag{1}$$ ... 0answers 25 views 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 ... 0answers 7 views special partition function Let following be the partition function in infinitely many variables$x_i$the linear coordinates on the vector space$V. $$\mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\... 1answer 19 views determine the stability of an equilibrium point(x,0). I would like to determine the stability of equilibrium point (x,0) of the differential equation \dot x = Ax A= \bigg[ \begin{matrix} 0&0\\0&a \end{matrix} \bigg] and a >0 ... 1answer 43 views proof (Legendre polynomial ) I'm stuck, couldn't figure it out. I appreciate your help. Show that :$$\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)}0answers 25 views 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}{... 3answers 61 views 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: (... 1answer 25 views 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 ... 0answers 28 views 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 ... 0answers 43 views 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 ... 0answers 24 views Find a solution bounded near x=0 of the following ODE Fine a solution bounded near x=0 of the following ODEx^2y''+xy'+( \lambda ^2x^2-1)y=0$$my attempt : this is Bessel's equation so let u=\lambda x then y(x)=y(\frac{u}{\lambda}) Also ... 0answers 16 views Solving a system of partial differential equation equivalent to finding integral curves? Consider the following system of partial differential equations defined on some neighborhood U of the origin in \mathbb{R}^2:$$z_x = F(x,y,z(x,y)), z_y = G(x,y,z(x,y))$$where F and G are ... 3answers 36 views Bernoulli equation with e^y term I'm having a hard time solving the following differential equation:$$(x^3 + e^y)y' = 3x^2$$I'm familiar with the approach of introducing z=y^{1-2}=y^{-1}, but that doesn't do the trick. Am I ... 1answer 20 views Discrepancy in solutions of differential equation? The differential equation at hand is this :$$ \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_{}^{}... 0answers 32 views Using Picard-Lindelof to find a solution toy'(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$such that the problem has a ... 2answers 95 views Integral of$\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$\...
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 $z$ ...
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 ...