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

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3answers
248 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
204 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
243 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
9k views

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
492 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
274 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
338 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 $s = -2-4i$ ...
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1answer
85 views

Solutions and attraction regions of following odes?

Assume a mapping $X: \mathbb{R} \to \mathbb{R}^d$. We know that the solution to ode $$ d X_t = (\mu - X_t) dt $$ is $X_t = (X_0-\mu) e^{- t} + \mu$, which indicates that $X_t$ converges to $\mu$ as ...
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2answers
1k views

How to apply reduction of order to find a 2nd linearly independent solution?

I have some questions about writing a general solution, $y$, for $y''-y=0$ when $y_1 = e^x$ is a known solution. I do not understand the logic of the method of reduction of order. How do we apply ...
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2answers
642 views

Using Octave to solve systems of two non-linear ODEs

How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: ...
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2answers
5k views

Difference between improper node and proper node for phase portrait

Can someone offer an explanation for the difference between these two? I see pictures of what seem to be examples of both, but it's hard for me to discern what a new portrait would be. Any help? ...
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1answer
295 views

Cubic population model question

Have the following population model, $$ \frac {dN}{dt}= cN(N-k)(1-N)$$ The first stage of the question is to investigate the steady states, however im a little stuck on finding the solution to the ...
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1answer
39 views

Proving ODE is periodic if and only if p(t) is periodic

Consider the first-order nonautonomous equation $x′ = p(t)x$ where $p(t)$ is differentiable and periodic with period $T$. Prove that all solutions of this equation are periodic with period $T$ if and ...
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0answers
75 views

How do I show that $y(t)=\int_0^tv(t-s)f(s) ds$ is a solution of $L[y]=f(t)$?

We have the $n$th-order scalar differential equation $$L[y]=\frac{d^{n}y}{dt^{n}}+a_1\frac{d^{n-1}y}{dt^{n-1}}+\dots+a_ny=f(t).$$ Let $v(t)$ be the solution of $L[y]=0$ which satisfies the initial ...
0
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1answer
36 views

how to construct a (2nd order) ODE that will be satisfied by a provided fundamental set?

If given a couple of functions and asked to construct an ODE of the form $y'' + q(x)y' + r(x)y = 0$ admitting of that couple as a fundamental set, once we've established that the couple could be a ...
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3answers
40 views

Converting a differential equation

Consider an ODE $\frac{dy}{dx}=h(x,y)$ such that $h(rx,ry)=h(x,y)$, which implies $h(x,y)=k\left(\frac{x}{y}\right)$. (Why?) Show that this ODE can be changed to a separable ODE for $u=u(x)$, ...
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3answers
185 views

Differential Equations Skydiver Problem

I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate. The famous differential equation problem models a skydiver jumping out of a ...
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2answers
49 views

How to solve this first-order differential equation?

I have been trying to solve a differential equation as a practice question for my test, but I am just unable to get the correct answer. Please have a look at the D.E: $dy/dx = 1/(3x+\sin(3y))$ My ...
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2answers
68 views

trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
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1answer
198 views

Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
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2answers
32 views

Find the first order system of linear equations

Regard the diff equation: $mϕ′′+aϕ′+(mg/L)ϕ=0$ $ϕ(0)=0.1$ $ϕ′(0)=0$ where $m=0.1,L=1,a=2,$ 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ...
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2answers
243 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
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1answer
38 views

Can someone give me an example of how to work out an exact linear second order differential equation?

I have a theorem that states: If an equation $P(x)y''+Q(x)y'+R(x)y=0$ can be written in the form: $$[P(x)y']'+[f(x)y]'=0$$ then the equation is said to be exact. Now I need to expand and equate ...
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2answers
82 views

Solve for $y' + Py = ae^{bt}$

How do I solve $y' + Py = ae^{bt}$? My attempt: $y' + Py = ae^{bt}\Rightarrow Py - ae^{bt} + 1.\frac{\mathrm{d} y}{\mathrm{d} t}=0$, where $M(t,y)=Py - ae^{bt}$ and $N(t,y)=1$. $M_{y}=P$, and ...
0
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1answer
266 views

Solving a first order linear ODE and determining the behavior of its solutions

(a) Draw a direction field for the given differential equation. How do solutions appear to behave as $t → 0$? Does the behavior depend on the choice of the initial value $a$? Let $a_{0}$ be the value ...
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1answer
32 views

Division of differential equations

$$\frac{dx(t)}{dy(t)}=\frac{\alpha x(t) - \beta x(t) y(t)}{-\gamma y(t) + \delta x(t)y(t)}$$ How would one simplify this fraction? Maybe the chain rule could be of any use, but I don't see how.
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2answers
86 views

Second order ODE - why the extra X for the solution?

Assuming I have the following homogeneous ODE equation: $$a\cdot y'' + b\cdot y' + c \cdot y = 0$$ Why for $(b^2 - 4\cdot a\cdot c=0) \quad $,(meaning, when $m_1=m_2$) then the solution is: $$y = ...
0
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1answer
134 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution ...
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1answer
104 views

Find the velocity of a flow

The question is: Find the velocity of the flow described by the velocity potential given in the polar coordinates $φ$$(r, θ)$ = $θ$, where $x = r cos θ$ and $y = r sin θ$, $r > 0, 0 ≤ θ < 2π$ ...
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1answer
155 views

How to go about solving this question on differentials?

A ring of a planet has an inner radius of approximately 52,000 km (measured from the center of the planet) and a radial width of 19 km. Use differentials to estimate the area of the ring. (Round ...
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1answer
64 views

Initial Value Problem

Please help solving the following initial value problem: $$y''-3y'+2y \; = \; 3e^{-x}-10 \cos {3x}; \;\;\; y(0)= 1, \;\;\; y'(0)=2 $$ I have been working at it and have been hitting a road block
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1answer
45 views

Poincare-Bendixson theorem contradiction help

Lets suppose p is asymptotically stable but not a singularity for the planar differential equation dx/dt=f(x), then for points x sufficiently closed to p we must have x(t) tends to p. so the limit set ...
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1answer
45 views

Derivative of a differential equation help??

Please can someone explain this to me in detail: if $y''+4y'+3y=14\cos(2x)$ and $z'''+4z''+3z'=-28\sin(2x)$ show that the $z=y+c$ where $c$ is a constant I know the second is the integral of the first ...
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1answer
38 views

Solve a differential equation and evaluate the solution at a particular value of independent variable

If $\frac{dy(x)}{dx}=(2-3i)y(x)$ where $i=\sqrt{-1}$, what is the value of $y(\pi)$?
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2answers
82 views

Differential Equations/IVP: $\frac{dy}{dt} = 4 - y^3$ and $y(-1)=2$

Transform the given initial value problem into an equivalent problem with the initial point at the origin. $$\cfrac {dy}{dt} = 4 - y^3 \\ y(-1)=2$$ I have no idea about how to solve it. Could you ...
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3answers
662 views

Integrating absolute value function

I'm working on a problem drawing phase plane diagrams in my applied mathematics course. I'm supposed to draw the phase line diagram of $x''+\vert x\vert=0.$ In the process, I get to the differential ...
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1answer
461 views

Wronskian-Differential Equations

The equations below are matrices: Consider the vectors $y^{(1)} (t)$=$\begin{pmatrix}t \\1 \end{pmatrix}$ and $y^{(2)}$ (t)=$\begin{pmatrix}t^2 \\2t \end{pmatrix}$ (a) Compute the Wronskian of ...
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2answers
473 views

Solve separable DE with integrating factor and homogeneous substitution

I just came out of test which asked to solve $$\frac{dy}{dx}=\frac{y}{x}$$ with $x,y>0$ in three ways: by separating the variables, using the substitution $y=vx$ and using an integrating factor. ...
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2answers
502 views

Find a particular solution of the differential equation $-3y''-2y'+y=3xe^x$

Using the method of undetermined coefficients. Guess $(Ax+B)e^x$ Plug into diff eq: $-3[(Ax+B)e^x]'' - 2[(Ax+B)e^x]' + (Ax+B)e^x = 3xe^x$ Wolfram alpha simplifies this to: $A(x-2)=e^x(4B+3x)$. ...
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1answer
81 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
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1answer
48 views

Differential Equations: solve the system

Solve the following system: $$dx/dt=-.2(y-2)$$ $$dy/dt=.8(x-2)$$ This is what I have so far, but I got stuck.. $$\begin{eqnarray} dx/dt&=&-.2y-.4\\ x'&=&-.2y-.4\\ ...
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1answer
296 views

$\frac{dy}{dx} = 3y^{2/3}$ general solution?

What's the general solution of $\frac{dy}{dx} = 3y^{2/3}$ ? Im pretty sure this is a separable equation, but I'm not sure how to go forward? Just multiply by $dx$ and $\frac{1}{3y^{2/3}}$ well then I ...
52
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0answers
582 views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
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2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
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3answers
8k views

Links between difference and differential equations?

Does there exist any correspondence between difference equations and differential equations? In particular, can one cast some classes of ODEs into difference equations or vice versa?
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8answers
2k views

What's so special about sine? (Concerning $y'' = -y$)

In an attempt to actually grok sine, I came across the $y''= -y$ definition. This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent ...
16
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1answer
438 views

Dual Bases: Finite VS Infinite Dimensional Spaces

Motivation I know that in finite dimensional spaces like a $n$ dimensional Euclidean space $\mathbb{E}^n$, for every basis $G=\{g_1,g_2,...,g_n\}$ we can define a dual basis $G'=\{g^1,g^2,...,g^n\}$ ...
13
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4answers
478 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
9
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6answers
5k views

How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...
14
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3answers
1k views

Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$

Find all functions satisfying $f(2x)=2f'(x)f(x)$ Under given condition, can't we find explicit solutions?