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

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How can I solve this Initial Value Problem using the Euler method?

My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a ...
4
votes
1answer
178 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
4
votes
3answers
337 views

Closed form solutions of $\ddot x(t)-x(t)^n=0$

Given the ODE: $$\ddot x(t)-x(t)=0$$ the solution is: $$x(t)=C_1\exp(-t)+C_2\exp(t)$$ If we square the $x(t)$ we have: $$\ddot x(t)-x(t)^2=0$$ and the solution is given by: $$x(t)=6\wp(t+C_1;0,C_2)$$ ...
4
votes
2answers
287 views

Formula for integration bounds of recursively defined polynomial sequence

We can recursively define a sequence of polynomials by $$P_0(x) := 1$$ and then with the definite integral $$P_n(x) := \int_{c_n}^x P_{n-1}(t) ~\mathrm dt$$ where the $c_n$ are to be chosen so ...
3
votes
1answer
41 views

what can I say about the solution $y(x)$ of the ODE?

Let $y:\mathbb R\to \mathbb R$ be differentiable and satisfy the ODE: $$\frac{dy}{dx} =f(y),x\in\mathbb R$$ $$y(0)=y(1)=0$$ where $f:\mathbb R\to \mathbb R$ is a Lipschitz continuous function. Then ...
3
votes
3answers
70 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
3
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3answers
116 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
3
votes
1answer
1k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
3
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1answer
4k views

How to Solve the Coupled Differential Equations?

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED $\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ I am at a loss as to ...
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1answer
165 views

Initial value problem $t\frac{dx}{dt}=x+\sqrt{t^2+x^2}$

Another question on ODEs, this time just wondering how I should start with this one. $$t\frac{dx}{dt}=x+\sqrt{t^2+x^2}, \qquad x(1)=0.$$ It looks like a linear ODE and so after playing around with it ...
3
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1answer
2k views

Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$

So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method. $$ x(1 - x)y'' - 3xy' - y = 0 ...
3
votes
2answers
212 views

Solving ODE with substitution

I have this as homework: $$(xy^2+y)dx+(x^2y-x)dy=0$$ I tried to solve it by substituting $z=xy+1$, but got the answer like $y=Cxe^{xy}$, which, I guess, is wrong. I tried to solve it couple of ...
3
votes
1answer
461 views

Two-Point boundary value problem

To solve ${d^2y \over dx^2} =f(x)$, $0<x<1$ with $y(0)=\alpha, y(1) = \beta$. We can get a finite difference approximation by taking $$\frac{y_{j+1}-2y_j+y_{j-1}}{h^2} =f_j \\\Rightarrow ...
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3answers
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Simple Harmonic Oscillator Solution

In Physics, the Simple Harmonic Oscillator is represented by the equation $d^2x/dt^2=-\omega^2x$ . By using the characteristic polynomial, you get solutions of the form $x(t)=Ae^{i\omega t} + ...
3
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1answer
655 views

Why does acceleration = $v\frac{dv}{dx}$

If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$ However I also understand $a=v \frac{dv}{dx}$. Can ...
2
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1answer
56 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
2
votes
1answer
44 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
2
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1answer
83 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
2
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1answer
121 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & ...
2
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1answer
164 views

Eigenvectors Trajectories

I got stuck with a problem while studying for a control systems exam. It goes as following: "Look at the picture of trajectories of a linear, time-invariant system with the form: ...
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2answers
55 views

second order DE using reduction of order

Any Hints / details on how to find a second solution for $$x^2y'' + xy' -4y=0?$$ $$y_1 = x^2 y_2$$ I need to use reduction of order thanks
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1answer
115 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
2
votes
2answers
110 views

Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
2
votes
2answers
132 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
2
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1answer
173 views

Legendre Equation Properties

Is there a nice way to derive, starting from the Legendre differential equation, the generating function, the recurrence relation, the Rodrigues differential form & the Schlafli integral form ...
2
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0answers
676 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
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0answers
136 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
2
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3answers
681 views

Express differential equations as system of first order equations

Express the differential equation $$y'''-6y''-y'+6y=0$$ as a system of first order equations i.e. a matrix equation of the form $$A(\vec x)'=0$$ where $$\vec x\text{ is the vector }\left[ ...
2
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2answers
501 views

Considering the linear system $Y'=AY$

What would be an equation that I can use when I compute the eigenpairs for the coefficient matrix $A.$
2
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0answers
70 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
2
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1answer
66 views

Differential Equation - $y'=|y|+1, y(0)=0$

The equation is $y'=|y|+1, y(0)=0$. Suppose $y$ is a solution on an interval $I$. Let $x\in I$. If $y(x)\ge 0$ then $$y'(x)=|y(x)|+1\iff y'(x)=y(x)+1\iff \frac{y'(x)}{y(x)+1}=1\\ \iff \ln ...
2
votes
1answer
540 views

butcher tableau runge kutta methods

Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me Write the Butcher Tableau for the 1-stage $\theta$ method: $$U^n ...
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2answers
53 views

Seemingly easy Ordinary Differential Equation

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
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1answer
37 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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1answer
38 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
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1answer
50 views

Modelling population with $\frac{dP}{dt}=P(\beta - \delta P)$

The population $P(t)$ of a biological species can be modelled by $$ \frac{dP}{dt}=P(\beta - \delta P) $$ subject to $P(0)=P_0$ where $\beta$ is the birth rate and $\delta$ is the death rate. ...
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2answers
49 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
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0answers
71 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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3answers
237 views

Tricky Separable Differential Equation

Please guide me: $y' + ay +b = 0$ (a not zero) is supposed to be separable and has solution $y = ce^{-ax} - \frac ba$ Here is my start to this problem: $\frac{dy}{dx} + ay = -b$ is as far as I can ...
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2answers
161 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
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2answers
216 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
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2answers
119 views

How to solve this differential equation system?

The following system is given: $$ \dot{x} = y + z \\ \dot{y} = x + z \\ \dot{z} = x + y $$ The first thing I did was to find out the eigenvalues. I found out, that -1 is a doubled and 2 a single ...
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0answers
142 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
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3answers
245 views

I.V.P $y'=\sin(e^{y}), y(0)=a$

Is the I.V.P: $$\begin{cases} \dfrac{dy}{dx}=\sin(e^{y})\\[8pt] y(0)=a \end{cases} \text{ where } a\in \mathbb{R}$$ a) Find the values ​​of $a$ for which $y(x, a)=0$ b) Prove that if $a=0$ then ...
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1answer
183 views

First order non linear Ordinary differential equations

Consider the first order differential equation $\displaystyle\frac{dy}{dt} = f(t,y)= -16t^{3}y^{2}$, with the inital condition $y(0)=1$ Estimate the lipschitz derivative for the differential ...
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2answers
238 views

how did he conclude that?integral

So the question is : Find all continuous functions such that $\displaystyle \int_{0}^{x} f(t) \, dt= ((f(x)^2)+C$. Now in the solution, it starts with this, clearly $f^2$ is differentiable at every ...
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2answers
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Solving differential equation $x^2y''-xy'+y=0, x>0$ with non-constant coefficients using characteristic equation?

Whenever you deal with non-constant coefficients you usually use Laplace transform to solve a given differential equation, at least that's how how I learned it. But how would you solve the equation ...
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3answers
337 views

Is $f'(x)-3f(x) = 0$ subspace of differentiable functions $f\colon (0,1)\to \mathbb{R}$

$V$ is space of differentiable functions $f(0,1) \to \mathbb{R}$ and $W$ is a subset of $f$ that meets $f'(x) - 3f(x) = 0$ for all $x\in (0,1).$ Is subset $W$ a subspace of $V$? I know that I have ...
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1answer
226 views

Multistep ODE Solvers

Write both a fourth order Adams Bashforth and Adams Moulton procedure to solve $$x'(t) = x(t)-y(t)-\exp(t);$$ $$y'(t) = x(t)+y(t)+2\exp(t)$$ with initial values $x(0) = -1, y(0) =- 1$ on the ...
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2answers
321 views

Second Order Differential Equations e times sin particular solution

The differential equation I am trying to solve is $$ \dfrac{d^2y}{dt^2} + 4\dfrac{dy}{dt} + 20y = e^{-2t}\sin(4t) $$ I know how to start off. I have done the $s^2 + 4s + 20 = 0$ to get $s = -2-4i$ ...