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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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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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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1answer
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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
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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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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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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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17 views

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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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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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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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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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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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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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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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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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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1answer
26 views

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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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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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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1answer
28 views

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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1answer
34 views

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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1answer
12 views

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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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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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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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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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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59 views

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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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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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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1answer
58 views

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
39 views

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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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 ...
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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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When integrating, can only one term of an equation be integrated or must entire equation be integrated to maintain equality?

Is integration considered a basic operation in the sense you have to do it to all parts of the equation? $y dy - x dx = 0$ Is it valid to do $\int y dy - \int x dx = \int 0$ but invalid to leave out ...
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How to put the equation $y'' + ky =0$ into Sturm-Liouville form?

I just wondering how do you put $$y'' + ky =0$$ into Sturm-Liouville form. Reason: I am trying to determine if the equation is Sturm-Liouville on the interval $[-3,4]$.
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1answer
28 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
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solve $y'=ay+b$ [duplicate]

I have this differential equation which I want to solve $\displaystyle\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$ I know it is in the form $y'=ay+b$ But how can I solve it ?
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Initial conditions of a transformed differential equation

I'm not sure if the title is appropriate but this is what I mean. Suppose $$ \frac{dy}{dt} = y(t),~~~ y(t_0) = y_0~~~~~~~~~~~~~(1)$$ and $$ \frac{dx}{dt} = x(t),~~~x(t_0) = x_0 ~~~~~~~~~~~~~~(2)$$ ...
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Why before $e^{x}$,the solution was not possible?

we know the important role of exponential function in solving of ordinary differential equation.But the solution can be done by using another function like $10^x$ or $2^x$.The example below shows ...
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Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?
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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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Solve DE with trasfer function

How would you solve $\frac{\mathrm{d} C}{\mathrm{d} t}=\frac{F}{V}C_{i}-\frac{F}{V}C-kC^{3}$ given $\frac{F}{V}=0.1, k=0.5,$ $\frac{\mathrm{d} C}{\mathrm{d}}=0$ ? I´m supposed to use a transfer ...
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1answer
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Why is it sometimes it seems like you can integrate with respect to x or y and treat the other as a constant, and other times you can't?

I am very confused right now. I thought we can't just do algebra on an ODE to find the solution. The following isn't allowed: $x dx + (y - 2x)dy = 0$ $x dx=-(y-2x)dy$ $\int x dx = \int -y+2x dy$ ...
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2answers
29 views

Getting wrong answer with exact equation with initial condition.

Solve the initial value problem $(4y+2t-5)dt+(6y+4t-1)dy=0, y(-1)=2$ This is an exact equation with $M(t,y)=\frac{\partial f}{\partial y}=4y+2t-5$ and $N(t,y)=\frac{\partial f}{\partial t}=6y+4t-1$ ...
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1answer
44 views

Differential Equation - Water evaporation

Given that a glass of water is filled to its fullest, $10\,cm$ in height, and that after three days the water level is at $9\,cm$ in height. Find when the glass will be empty. The water is ...
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Locally Linear Systems-repeated $\lambda$

For a locally system whose corresponding linear system has repeated eigenvalues, the type of equilibrium point cannot be determined. I know that the locally Linear system equilibrium can possibly be a ...