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

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Partial Differential Equations Course And Differential Geometry Prerequisites

Is the ordinary differential equations course a prerequisite for the partial differential equations course for a person who has passed the integral calculus course? Is it really required to have ...
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How to solve the differential equation $(2xy^2-y){dx}+(y^2+x+y){dy}=0$? [on hold]

I'm weak at solving equations like this: $$(2xy^2-y){dx}+(y^2+x+y){dy}=0$$ Please show how to complete equation, so that it becomes exact. Thank you for help in advance.
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Linear ODE and Fourier Series

Let $m,k_0,k$ be positive real numbers and $x_1$, $x_2$ be real-valued functions of time. Suppose we have following system of two coupled ODEs ( motivated by a coupled oscillator with two masses ...
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1answer
51 views

McLaurin series expansion to evaluate a function

I have a maths assignment due for college based on the McLaurin series and don't understand how to do it. I need to use a McLaurin series expansion to evaluate a function. The function is the ...
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2answers
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Solving the integral equation $y(x) = 3 + 2\int_1^x t y(t) dt $ by reducing it to a differential equation

Solve the integral equation $$y(x) = 3 + 2\int_1^x t \ y(t) \ dt $$ First I solved for the integral equation. Then I'm told to differentiate and I get $${dy \over dx} = 2 x y(x) $$ Then I ...
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Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
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36 views
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Linear systems of differential equations

I would like to see an example of a real physical situation where one can find a set of variables evolving according to a system of linear differential equations. I wasn't able to find any such ...
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1answer
29 views

How to solve ODE's $\dot{x}=ax+by$ and $\dot{y}=bx+cy$?

I need help in solving a system of ODE's $$x'(t)=ax(t)+by(t) \mbox{ and } y'(t)=bx(t)+cy(t)$$ where $a,b \in \mathbb{R}$ and $x,y$ denote standard co-ordinates in $\mathbb{R}^2$. I checked on ...
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2answers
31 views

Can an arbitrary constant in the solution of a differential equation really take on any value?

Consider the first order differential equation $y' = -2y^{\frac{3}{2}}$. It has $y = \dfrac{1}{(x+c)^2}$ as the solution. Now, if I divide both the numerator and denominator by $c^2$ (assuming $c ...
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Existence & uniqueness of a second order ODE

$(x+d)\ddot{x}=gh+\frac{P_{o}}{\rho}-\frac{P_{o}}{\rho}\left(\frac{L}{L-x}\right)^{\gamma}-sgn(\dot{x})\frac{f}{2D}x\dot{x}^{2}-\frac{1}{2}\dot{x}^{2}, x(0)=\dot{x}(0)=0$. Here $g, h, L, \rho, P_{o}, ...
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Convert the following system to a first order system:

Really having a hard time with this.....Convert the following system to a first order system: $$\frac{d^2x}{dt^2} -3\frac{dy}{dt}+x=\sin(t)\\ \frac{d^2y}{dt^2} -t\frac{dx}{dt} - ye^t =t^2$$
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Differential equation to space state excercise

This is a "back of chapter" excercise which im trying to solve, my answer doesnt match the solution printed on the book, I want to write the equation in state space matrix form without using the ...
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41 views

How to solve $4x^2\cos y\sin y\partial{y}-3x\sin y\partial{x}+8\sin^2y\partial{y}=0$?

$$4x^2\cos y\sin ydy-3x\sin ydx+8\sin^2ydy=0$$ find the solution of this Bernoulli equation. How can I start?
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1answer
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periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$

Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$ I want to discusse about non-constant periodic solution of it. Can someone give a hint that how to start to think. And does it have ...
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Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
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267 views

Estimating a dynamical system's behavior without using Liapunov theorem

Assume that we have the following dynamical system $$x'=(\epsilon x+2y)(1+z)$$ $$y'=(-x+\epsilon y)(1+z)$$ $$z'=-z^3$$ Then how can I show that any solution that started from the region $z>-1$ ...
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Solve the given initial value problem.I need your help.

Solve the $$x'=tx^2+x-t^3\,,\quad x\left(\, 2\,\right)=1$$ I need its exact solution not a numerical solution.In fact I have to compare the exact solution with the numerical solution.I tried it but I ...
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homogenous differential equation with variable coefficient [on hold]

Please help to find solution of boundary value problem $$ y''+xy=0 $$ $x \in [a,b]$ with $y(a)=y(b)=0$
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Help with first order linear PDE with initial condition

I would like to solve the following pde: $2y\cdot \partial_x u(x,y)-3x\cdot\partial_yu(x,y)=0$ and $u(x,x)=e^{x^2}$ Without the initial condition I got the following result: ...
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Phase Plane Analysis

Classify the fixed point at the origin and sketch an accurate phase portrait for the following system: $$\left\{\begin{matrix} \dfrac{dx}{dt}=36x-16y\\ \dfrac{dy}{dx}=-3x+28y \end{matrix}\right.$$ ...
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Practical applications of first order exact ODE?

In elementary ODE textbooks, an early chapter is usually dedicated to first order equations. It is very common to see individual sections dedicated to separable equations, exact equations, and general ...
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What are the ordinary and singular points of the first order diff. equation?

Consider a first order differential equation. What do ordinary and singular points mean? What do they represent? (I cannot understand their formal definitions so please explain with examples. Thank ...
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Equilibrium points and linear stability

Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, ...
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262 views

On a linear 3x3 system of differential equations with repeated eigenvalues.

I have the following system: $$\begin{cases} x'= 2x + 2y -3z \\ y' = 5x + 1y -5z \\ z' = -3x + 4y \end{cases} $$ $$\det(A - \lambda I)= -(\lambda - 1)^3$$ the eigenvector for my single eigenvalue ...
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indicial equation of a differential equation

The indicial equation for $x(1+x^2)y'' + (cosx)y' + (x^2-3x+1)y=0$ is $r^2=0$. How it is possible. I reduced the given diff eqn as: $x^2y'' + \frac{xcosx}{1+x^2}y' + \frac{x(x^2-3x+1)}{1+x^2}y=0$. ...
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1answer
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Differential equation: $A(x)y''(x)+A'(x)y'(x)+y(x)/A(x)=0$

So give the differential equation $$A(x)y''(x)+A'(x)y'(x)+\frac{y(x)}{A(x)}=0,$$ with $A(x)$ a known function and $y(x)$ te be determined. What is the solution for this differential equation ? I've ...
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How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
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a question regarding wronskian

I was working on following problem: Let $y_1$ and $y_2$ be solutions of $x^2y'' + y' + (\sin x)y = 0$ satisfying $y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $. I worked like following: since ...
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Limit to infinity from a differential equation

Let $R'(t) + \nu R(t) = \nu F(t)$, $F(0)=0$, $R(0)=0$, $f(t) \geq 0$, $F(t) = \int_0^t f(\tau)d\tau$, $F(t) \leq 1$, and $\lim_{t \rightarrow \infty} F(t) = 1$. I solved the differential equation and ...
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The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
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1answer
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example which doest not satify Lipchitz condition but has unique solution

$y'=1+\sqrt y , y(0)=0 $ Show that this IVP does not satify Lipchitz condition but has a unique solution. I have shown the first way, like this: Let $f(x,y)=1+\sqrt y $.Then $\frac ...
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50 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?
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Can the Heat Equation be Averaged Over a Region?

I am doing a project for my partial differential equations class in which I am motivating the definition of a weak solution. To get started, I assumed that $T$ was a solution to $\nabla^2 T = \partial ...
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1answer
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How do we know radioactive decay can be modeled by the half-life equation, dq/dt = -aq?

I understand how to solve it. but why does $$\frac{d \lambda}{dt} = -k \lambda$$ The equation, in and of itself, means the rate of decay is proportional to the amount at a given time. How do we know ...
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Showing unstablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
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What is the difference between single and double modulus signs. Do they both mean magnitude?

What's the difference between a set of single modulus and a set of double modulus signs? On textbooks I have seen the magnitude of two vectors vector as $|\mathbf{x} - \mathbf{y}|$ but I've seen ...
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How to separate variables in this equation: $\;y\frac{dy}{dx} = (x+7)(y^2+6)\;?$

I need to solve the differential equation $$y\frac{dy}{dx} = (x+7)(y^2+6)$$ I know that the first step is to isolate both term each side and then integrate... But I can't figure out how to isolate ...
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1answer
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Differential equations - maximal domain

I was solving an exercise about differential equations, and i really don't get how can I determinate the maximal domain of solution. Example: $$(dy/dx) = x - y/(1+x), y(0) =-1$$ The solution is ...
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Inhomogeneous ODE (2nd order) - question to Laplace-transformation?

I've the following inhomogeneous second order ODE: $$a_1\cdot u(t) + a_2\cdot u'(t) + a_3\cdot u''(t) = b_1\cdot y(t) + b_2\cdot y'(t) + b_3\cdot y''(t)$$ The parameters $a_i$ and $b_i$ are ...
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Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
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1answer
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mass-spring system. what is y(t)? [on hold]

Consider a mass-spring system with unit mass (m = 1), spring constant k = 9, critically damped, and no external force. Suppose that the oscillator starts at rest, and slightly compressed, at the point ...
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After my first question I tried to solve the equation $\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$

After my first question I tried to solve my differential equation $$\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$$ Here is what I have done until now. I used $y'=b-a \cdot y$ and the ...
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381 views

Differential Equations and Newtons method

How can I approach this question? For problem one this is what I did: Given the DE, $$p'(x) = p''(x) + \left(2\pi*\frac{f}{c}\right)^2p(x) = 0,$$ and its solution, $p(x) = \sin(kx)$, I substituted ...
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1answer
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Proving inequality $(x^2+y^2)(y-1)+yx-y^2<0$

I have an inequality which came out of Lyapunov function for system of ODE's: $$(x^2+y^2)(y-1)+yx-y^2<0.$$ To prove stability of my solution, I have to prove that the inequalty is true in area ...
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1answer
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How to express a system of differential equations in a form suitable for numerical methods?

I am modeling rocket thrust equations using some of the formulas and derivations on page 37 & 38 here. For my Rocket model, I have the following two equations: $$dv/dt = 383v^2$$ $$dA/dt = 635.14 ...
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continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
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1answer
32 views

Solve Basic partial differntial equation question [on hold]

Help me solve the partial differential equation. $$\frac{\partial^2z}{\partial x^2} + 2 \frac{\partial^2z}{\partial y^2} - 3\frac{\partial^2 z }{\partial x \partial y} = e^{2x-y} + e^{x+y} + \cos(x ...
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Logistic model. Did I set up the differential equation $(1)$ correctly?

Update: I fixed it. The major mistake I made was that originally put $I(t) = \beta\cdot(P-y(t))$ while it of course is supposed to be $I(t) = \beta\cdot y(t)$. NB: I came up with this problem ...
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$\Phi(t)=P(t)e^{tR}$ as a fundamental set for $x''(t)=\sin(t)x'(t)$

Problem. Find $2\times2$ matrices $R$ and $P(t)$ such that $R$ is constant, $P(t)$ is periodic, and $\Phi(t)=P(t)e^{tR}$ is a fundamental set of solutions for $x''(t)=\sin(t)x'(t)$. $ $ Attempt at ...
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Existence And Uniqueness Theorem Question

(a) Does the existence and uniqueness theorem guarantee the uniqueness of the solution of the initial value problem $dy/dx = 2x(y-2)^\frac{2}{3}, y(1) = 2$ Attempt: NO because $∂/∂y = \frac{4x}{3 ...