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

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6
votes
2answers
165 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
0
votes
2answers
223 views

find equilibrium points for an infected population IVP

Let $x$ be the proportion of a population with a disease, let $y$ be the healthy ones the disease rate spreads at $$\frac{dy}{dt} = ay(1-y),\quad y(0)=y_0$$ $(a)$ find the equilibrium points for ...
1
vote
2answers
54 views

logistic equation IVP doubled

Suppose a population follows the equation $\frac{dy}{dt} = ry[1-\frac{y}{K}]$ (let $y_i =$ initial value of $y$) a.) $y(0) = \frac{K}{3}$ find the time $T$ at which the initial population has ...
0
votes
1answer
79 views

How have they calculated this maximum point of the Verhulst Model

The simplest guess is using the Vehulst model: $r = r_0( 1 - N/K)$. Then the dynamic equation is $\frac{dN}{dt} = r_0N(1 - \frac{N}{K})$. From here, we can calculate the maximum point to be at ...
2
votes
0answers
40 views

a nonlinear ode with an quotient

I'm curious about the nonlinear ODEs with an quotient including dependent variable : $$y''(x)+\frac{Ay(x)x}{1+y(x)}+Bx=0$$ Could you give me a clue on solving this equation explicitly?
1
vote
2answers
94 views

non-linear ordinary differential equation

Studying some Newtonian mechanics, I've encountered this differential equation : $y'+a y^2=b$ where $a,b$ are constants. how could we solve it ? (I trying to get an algebraic solution)
1
vote
2answers
2k views

Find the equation of the tangent to the curve with exponential function

The question is as follows: Find the equation of the tangent to the curve $y = xe^{2x}$ at the point $(\frac{1}{2}, \frac{e}{2})$. Now I figured out that $\frac{dy}{dx} = e^{2x}(2x+1)$, and that ...
1
vote
0answers
110 views

How can I solve this system of equation analytically?

$$ \begin{align} i x'(t) &= c A(t)(e^{-i q(t)}) y(t)\\ i y'(t) &= c A(t)(e^{i q(t)}) x(t)\\ \\ A(t)&=a cos(wt-r)\\ q(t)&=b+dt+k sin(wt+r) \end{align} $$ where all of $a,b,c,d,k,w,r$ ...
1
vote
0answers
159 views

Analytic solution at an irregular singular point of linear ODE

It is easy to construct examples of linear ODEs with a regular singular point $z_0$ and a regular at $z_0$ solution. For instance, if one of the characteristic exponents at $z_0$ is a nonnegative ...
11
votes
4answers
335 views

How to prove that $\frac{d^n}{dx^n}(x^2-1)^n=0$ has $n$ real roots?

How do I prove that $$\frac{d^n}{dx^n}(x^2-1)^n=0$$ has $n$ real roots?
6
votes
3answers
271 views

Existence of an extremum for the solution of the ODE $\ddot{x}+\frac32x^2=0$

Consider the 2nd order ODE $$ \ddot{x}+\frac32x^2=0. $$ Denote by $u$ the maximal solution of the associated Cauchy problem with initial condition $(x(0),\dot{x}(0))=(0,1)$. The problem is to prove ...
3
votes
1answer
213 views

Center manifold of sets of equilibria

My question is regarding Center Manifolds containing a continuous set of equilibrium points. The theory I have studied talks about the existence of a center manifold for equilibrium points, but what ...
0
votes
1answer
184 views

Solving wave eigenmodes in a cylinder

I am trying to find the eigenfrequencies of waves in a cylinder, or put into equations: $$\frac{1}{c^2}\frac{\partial^2}{\partial t^2}u = \Delta u$$ with $u = u(t,r,\theta,z)$. Giving, in cylindrical ...
2
votes
2answers
400 views

Existence and Uniqueness of solutions

I'm pretty confused about this topic in Differential Equations. It's a simple topic, but I just can't get the gist of it. Here are two examples that I would like for you to explain to me ...
1
vote
1answer
470 views

Determining if a function satisfies a Lipschitz condition

I have $f(t, y) = e^{t - y}$. I want to see if $f$ satisfies a Lipschitz condition on $D = \{(t, y): 0 \le t \le 1, \; - \infty < y < \infty \}$. I am using this definition: A function ...
1
vote
2answers
1k views

Diff eq. transformation polar coordinates

I have $(x',y')=(x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}},x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}} )$ Now I want to use polar coordinates $(x,y)=(r\cos(t),r\sin(t))$ to get ...
3
votes
3answers
155 views

prove: if $y=\frac{dy}{dx}$ then , $y=ce^x$ for some constant $c$

we all know that : if $y=c e^x$ then $ y= \frac{dy}{dx}$ let $y=f(x)$ now , we want to prove the other way, I mean : prove,if $y=\frac{dy}{dx}$ then , $y=ce^x$ for some constant $c$ can any ...
0
votes
3answers
62 views

Finding a particular solution to $y'- \frac{3}{2}y=3t+2e^{t}$

I given that $y(0)=y_o$ and I have to find the value of $y_o$ that separates solutions that grow positively as $t \rightarrow \infty$ and that grow negatively. How do I that? I already found the ...
0
votes
1answer
170 views

Poincaré map corresponding to submanifold

I am interested in how to calculate the Poincare map corresponding to a submanifold. The vector field $f(x,y)=(-y,x)$ has the periodic solution $(cos(t),sin(t))$ Now I'd like to compute the Poincare ...
3
votes
2answers
86 views

Let $y(t)=(y_1,y_2)^t$ satisfy $\frac{dy}{dt}=Ay,t>0$ such that..

I came across the above problem and my attempts are as follows: We see the characteristic polynomial of $A$ is $t^2-{\rm trace} (A)t+\det(A)=0$ and hence $t^2- \lambda^2=0$ since ${\rm trace}(A)=0$ ...
1
vote
0answers
226 views

Cylindrical waves

I am trying to solve the general equation for cylindrical symmetric waves: $$\frac1{c^2}\frac{\partial^2u}{\partial t^2}= \frac1r\frac{\partial}{\partial r}(r\frac{\partial}{\partial r}u)$$ with $u = ...
2
votes
2answers
91 views

The solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=2xy^2$

Question from pg 32 of Barron's AP Calculus The solution of the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=2xy^2$ for which $y = -1$ when $x = 1$ is (A) $y = -\frac{1}{x^2}$ for ...
1
vote
1answer
83 views

Solve the DE $yy^{\prime}+x=\sqrt{x^2+y^2}$

Solve the DE $$yy^{\prime}+x=\sqrt{x^2+y^2},x>0$$ I show it is a homogenous first order DE and I use the substitution $y=Vx$. Then, I solve until $$\int \frac{V}{\sqrt{1+V^2}-(1+V^2)}dV=\int ...
1
vote
0answers
93 views

Half-stable fixed point on a circle

On a line graph, it's clear that a half-stable fixed point is the limit of moving the unstable fixed point towards the stable fixed point. Some solutions go to infinity depending on the initial ...
2
votes
1answer
68 views

DE involving Hyperbolic Trig function (Quick check)

I'm working problem 15 on page 133 of Boyce/DiPrima's Elementary Differential Equations and Boundary Value Problems (10th ed.) Note: this is homework, but it is not graded/turned in. I have arrived ...
1
vote
1answer
42 views

Notation issue regarding differential equations

I am given the following problem: Find a basis of solutions for the equation: $u^{iv} + 2u'' + 3u = 0$ The notation is an exact duplicaticate of what our professor used in his notes. Does anybody ...
1
vote
1answer
478 views

System of differential equations with triple eigenvalue

Could anybody, please, explain to me, how to solve system of 3 differential equations, when it has triple eigenvalue? I mean... we solved these equations by creating a matrix $A$ of the system and ...
1
vote
0answers
55 views

What's the shooting algorithm for the mass-spring problem (ode)?

I have the following problem : $$ \begin{aligned} \frac{d x(t)}{dt} &= y(t)\\ \frac{d y(t)}{dt} &= -x(t)+y(t)(1-x(t)^2)+u(t) \end{aligned} $$ with the initial condition $(x,y) = (0,0)$. Those ...
2
votes
2answers
168 views

general solution to $u'=\left(\begin{matrix} -1 & 1\\ 0 & -1 \end{matrix}\right)u$.

How to find the general solution to the following linear system: $$\left(\begin{matrix} u_1'\\ u_2' \end{matrix}\right)=\left(\begin{matrix} -1 & 1\\ 0 & -1 \end{matrix}\right) ...
1
vote
0answers
232 views

Use the existence and uniqueness theorem to prove a solution of DE

Use the existence and uniqueness theorem to prove that $y(x)=3$ is the only solution to the IVP $$y^{\prime}=\frac{x}{x^2+1}(y^2-9), \,y(0)=3$$ I have no idea on how to start. How to show the ...
3
votes
2answers
130 views

Solve $y^{\prime \prime}-(y^{\prime})^2-y^{\prime}=0$

Solve $y^{\prime \prime}-(y^{\prime})^2-y^{\prime}=0$. I use $$u=\frac{dy}{dx}$$ to transform the DE into $$\frac{du}{dx}-u^2-u=0$$. I know that this is an Bernoulli equation with $n=2$. I get the ...
1
vote
3answers
568 views

differential equation : non-homogeneous solution, finding YP

hi i have a problem for this Differential Equations : $$ \frac{d^{3}y}{dx^3} - 9\frac{dy}{dx} = 10 - 4x $$ i know first we must solve the homogeneous equation: and my result is : $C_1 + C_2e^{3x} + ...
0
votes
1answer
299 views

Consider the differential equation $dy/dt=ay-b$

Let $Y(t)=y-y_e$ ($y_e $ is equilibrium). Thus $Y(t)$ is the deviation from the equilibrium solution. Find the differential equation satisfied by $Y(t)$. I solved for the general solution and got ...
3
votes
3answers
396 views

Given the differential equation $(1+t^2)dy/dt+4ty=(1+t^2)^{-2}$

So I have to use the method of integrating factors. I thought I could do $${d\over dt}[(1+t^2)y]=2ty+(1+t^2)$$ But that doesn't give me the $4ty$ that I need. Help?
2
votes
2answers
351 views

Differential Equations: Population Problem $dp/dt= 0.5p - 380$

I just want to make sure this is right because I'm doing the homework online and I'm on my last attempt and I'm pretty sure I got the other two right yet the computer program said no. First at I ...
1
vote
2answers
577 views

Determine if a system described by a differential equation is linear

A system ("A System is any physical set of components that takes a signal, and produces a signal") is described by this equation: $ \frac{dy(t)}{dt} + 3 \times y(t) = x(t) $ Where $x(t)$ is the ...
4
votes
1answer
109 views

How do I squeeze a $\theta(t)$ and $\varphi(t)$ out of this?

A ball attached to a fixed-length massless rod swings about under gravity. Mathematically: $$L=T-U=\frac{MR^2}{2}(\sin^2(\theta)\dot{\varphi}^2+\dot{\theta}^2)+MgR \cos(\theta)$$ ...
4
votes
1answer
37 views

Which solution does a particle follow if uniqueness fails in ODE?

If there are multiple solutions to an ODE at y(0) = 0 and a particle is "dropped" into the flow field where the solutions intersect,which path does the particle take? In other words, which solution ...
1
vote
0answers
55 views

Neglecting solutions and reforming the system of differential equations with reducing the order but to keep choosen solutions

Here I have one problem which should help me to understand how to transform the system of differential equations with the condition to neglect two of four solutions and to get the appropriate system ...
2
votes
1answer
87 views

Solving Simple Mixing Problem

The Question: Assume there are two tanks, Tank A and Tank B. Assume Tank A contains 6 gallons of liquid and Tank B contains 9 gallons of liquid. Initially, Tank A contains $A_0$ pounds of salt and ...
5
votes
2answers
1k views

How to solve exact equations by integrating factors?

I know how to solve an exact equation like $$M(x,y) + N(x,y)y=0 $$ We just check $$\frac{\partial M}{\partial y} =\frac{\partial N}{\partial x} $$ If so, then it's just a little bit of algebra, ...
0
votes
1answer
360 views

Solve a first order differential equation by substitution

For a given differential equation $$xy'+2y\log y -4x^2y = 0;$$ $$ y(1)=1 $$ I want to use the substitution $v=\log y$. Which implies that $$v'=\frac{dv}{dy}\frac{dy}{dx}= \frac{1}{y} y'$$Hence: ...
1
vote
3answers
57 views

$y'=(8\cos8x)/(3+2y)$ with $y(0)=-1$ Initial Value Problem: DiffEq

Im told to find the explicit form $y(x)$ from the given differential equation and its initial value. Then find where the solution $x=?$ attain a maximum. What I did was: $3+2ydy=8\cos 8xdx$ ...
2
votes
0answers
175 views

Condition so that $y''+p(x)y'+q(x)y=0$ can be converted in a ODE with constant coefficients

I have to find a necessary and a sufficient condition for the functions $p$ and $q$ so that the linear differential equation : $y''+p(x)y'+q(x)y=0$ can be converted in a linear differential equation ...
6
votes
1answer
232 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
1
vote
1answer
194 views

The differential equation $dy/dx = 60 (y^2) ^ {1/5} $; $x>0$ $y(0)=0$ has how many solutions?

The differential equation $dy/dx = 60 (y^2) ^ {1/5} $; $x>0$ $y(0)=0$ has (1) A unique solution. (2) Two solutions. (3) No solution. (4) Infinite number of solutions. After solving I get ...
2
votes
2answers
188 views

find all solution to the equation $y'=Ay+b(x)$ for given $A$ and $b(x)$

I am asked to find all solution to the equation: $$y'= \left( \begin{array}{cc} 13 & 12 \\ 12 & 13 \end{array} \right) y+ \left(\begin{array}{c} x\\ 0 \end{array} \right)$$ No initial ...
0
votes
3answers
78 views

Differential Equation Basic - please explain the detail of this step

I'm looking through a solution of some problem. It has this step I don't quite understand. Please help me clarify. Relevant equations: $ u(x) = y^3 $ $\frac{dy}{dx} = \frac{1}{3y^2} \frac{du}{dx} $ ...
2
votes
2answers
605 views

Differential Equation: Using substitution $u(x)=y+x$, solve $\frac{dy}{dx} = (y+x)^2$

$u(x)=y+x$, solve $\frac{dy}{dx} = (y+x)^2$ I'm unsure about what dy/dx becomes. Thanks.
1
vote
1answer
83 views

If $f(cx, cy) = f(x, y)$ can we always find $g$ such that $g(\frac{x}{y}) = f(x, y)$?

Motivation: A differential equation $\frac{dy}{dx} = f(x, y)$ is said to be homogeneous if we can find $g$ such that $g(\frac{x}{y}) = f(x, y)$, which allows us to solve the equation using the ...