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

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7
votes
3answers
142 views

For what functions is $y'' = y$?

What functions $y = f(x)$ have the property that $f(x) = f''(x)$, i.e. what functions have the same integral and derivitive? I could think of $ce^x$ and $ce^{-x}$ (where $c$ is a constant), but are ...
0
votes
0answers
11 views

Why would you choose the Method of Frobenius over a Power Series solution to solve a DE?

I'm trying to determine where it would be more appropriate to use one or the other. To further clarify: Where would it make more sense to use: $y=\sum_{n=0}^{\infty}c_n(x-x_0)^{n+r}$ instead of ...
0
votes
1answer
18 views

Integrating a second order non homogeneous ODE

I took an exam and the teacher didn't solve this problem during the correction. I need to solve $$y''(x)-y(x)=\sin (e^x)$$I was able to find the solution to the homogeneous equation ...
0
votes
0answers
14 views

A spherical balloon is being inflated. Find the approximate change in volume if the radius increases from 5.9cm to 6 cm.

How do you use $\frac{dV}{dr}$ to solve this equation. Do you set $5.9$ equal to $\mathbb R$ and 6 equat to dr?
0
votes
1answer
66 views

Wave Equation Partial Differential EEquation

Basically I got a simple wave equation with an extra twist. The PDE is $\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L $ with homogeneous boundary condition As usual, ...
0
votes
0answers
31 views

Possible to proof/disproof this statement?

Given 2 different function $E=f(p,v)$ $\frac{dE}{dv}=g(p,v)$. $E=f(p,v)$ $\frac{dE}{dv}=f_p(\frac{dp}{dv})+f_v=g$ $\frac{dp}{dv}=\frac{g-f_p}{f_v}=x(p,v)$ From this formula got the ...
-3
votes
2answers
62 views

How many rabbits left [on hold]

The number of rabbits in a farm increases at a rate proportional to the number of rabbits at a certain time. The number of rabbits doubled to 10000 from the beginning of the year 1985 until the ...
1
vote
1answer
21 views

First order non-linear ordinary differential equation

The following ODE is given: $a\pm\sqrt{b+c*(x(t)+d))}=e*x'(t)+f*x(t) $ from Matlab I'm able to get a solution for the differential equation (actually two solutions, one for the + and one for the - ...
1
vote
2answers
20 views

Solving second order linear differential equations with non-constant coefficients

Can any one tell me what is the general method to solve the second order differential equation like this: $$ t(t + 1) y '' + (2 - t^2) y ' - (2 + t) y = (t +1 )^2$$ If the general method is ...
0
votes
2answers
48 views

Need a textbook for math course

The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and ...
-1
votes
0answers
18 views

Adomian method (how was the solution in this problem obtained?) [on hold]

can someone please help explain to me how the y terms in problem 1 of this paper were obtained in detail paper: http://www.ccsenet.org/journal/index.php/jmr/article/view/45923/24853 thanks here is ...
1
vote
1answer
65 views

Numerical series

Consider the series below that consist of 2 different formula $P_aV_a^{1.4}=P_bV_b^{1.4} $ and $P_aV_a=P_bV_b$ that keep repeating itself in the whole sequence. By assuming $P_1$ and $V_1$ both=1, ...
0
votes
0answers
23 views

Second order linear ODE and undamped

I am a bit confused with this problem: An object with mass 1 slug is attached to a vertical coil spring of spring constant of 1 pound per foot. After coming to equilibrium, the object is set into ...
3
votes
3answers
54 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 $a_1 = 0$ ...
-2
votes
0answers
15 views

differential equation spring equation and solve [on hold]

A mass is suspended from a spring. At rest, the mass 3kg stretches the spring .2 meters from its natural length. The mass is then pulled down .4 meters below the equilibrium position and released. ...
1
vote
0answers
16 views

find particular solution $y''+y'+4y=t^2+(2t+3)(1+cos(t))$

find particular solution $y''+y'+4y=t^2+(2t+3)(1+Cos(t))$ I'm looking for a way to make the right side less complicated or if there is any trick. What I would do is break it into two parts and add ...
2
votes
0answers
29 views
+50

What does it mean to extend curves to the projective line

The question is (from Arnol'd 's book : Ordinary Differential Equations): "Which of the differential equations $ \dot x = x^ n$ determine on an affine line a phase velocity field that can be extended ...
0
votes
1answer
22 views

Concept of inserting ansatz - separation of variables

In my textbook it says write the unknown function of two variables as a product of two functions of a single variable u(x, t) = X(x) T (t) but then the second step it goes straight away to have T ...
8
votes
3answers
279 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
0
votes
0answers
23 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
0
votes
0answers
59 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
0
votes
0answers
13 views

Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
0
votes
1answer
26 views

Equilibrium and Stability of Nonlinear Interactions

Examine the nonlinear model: $$\triangle x_t = rx_t(1-\frac{x_t}{K})-sx_ty_t$$ $$\triangle y_t = -dy_t+\epsilon x_ty_t$$ Find the equilibrium and their stability. Here all the parameters are ...
0
votes
0answers
19 views

Homogeneous Dilation of the Domain in the Free Membrane Problem

Consider the Neumann boundary value problem of the Laplace operator: $$ \begin{cases} \Delta u+\mu u=0,&\text{in }D,\\ \frac{\partial u}{\partial n}=0,&\text{on }\partial D. \end{cases} $$ Let ...
-1
votes
0answers
10 views

Solving Partial Differntial Equation with time-dependent Boundary Condition

How does one go about solving differential equations with a time-dependent boundary condition? For example something easy like: \begin{align*} \frac{dh}{dt} &=\frac{d^2h}{dx^2} \\ h(0,t) ...
2
votes
0answers
37 views

General solution of $ty'+2y=4t^2$

Should we left the general solution of the differential equation $t\frac{dy}{dx}+2y=4t^2$ as $t^2y=t^4+c$ instead of $y=t^2+c/(t^2)$ ($c$ is an arbitrary constant)? Does the solution $y=t^2+c/(t^2)$ ...
2
votes
0answers
22 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
-3
votes
0answers
15 views

Convolution - Laplace transform [on hold]

A friend of mine asked me for help. He needs these three exercizes about convolution: $$ \int\,\dfrac{ds}{s(s-1)} $$ $$ \int\,e^{3t} * \sin\,(5t)\,dt $$ $$ \int\,8t^2 * e^{8t}\,dt $$ When I ...
2
votes
3answers
37 views

Real analysis: simple second order ODE

I'm studying real analysis at the moment (just covered the mean value theorem, constancy theorem, applications to DEs etc.) and have run across this question that I'm stuck on. Any help would be much ...
2
votes
2answers
32 views

How to prove linearity?

Let suppose third-order differential equation, that solved for highest derrivative admits solution: $Y(t) = y(t) + C_1 f_1(t) + C_2 f_2(t) + C_3 f_3(t),$ where $y(t)$ is some solution, $f_1(t), ...
1
vote
1answer
25 views

Show that all solutions remain in the interval for all time

I really have no idea on how to get started with these, there's no similar example in my book. Do I need to compute $\frac{dy}{dx}$? Any help would be greatly appreciated. Maybe there's just some ...
2
votes
1answer
19 views

differential equation power series solution

I am trying to solve this equation using power series $$ (1-x)y"-xy'+y=0 $$ Knowing that $y(0)=-2$ and $y'(0)=6$. Please I need someone's help, I get a relation between $c(n)$,$c(n+1)$, and ...
1
vote
1answer
624 views

Solving Wave Equations with different Boundary Conditions

Right now I'm studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)...$) I know how to solve it when the boundary ...
0
votes
2answers
22 views

Clarification about correct steps to follow for second order ODE

I have to solve this Cauchy's Problem: $$\begin{cases}y''-y'+3y=x^2-x+3\\y(0)=y'(0)=0 \end{cases}$$ But I have a doubt about the correct steps to follow. It was told me that a second-order ODE is ...
2
votes
1answer
337 views

Existence and uniqueness theorems for ODE. Log-Lipschitz regularity.

Let $\mathbb{X}$ be a linear space with a complete metric $d:\mathbb{X}\times\mathbb{X}\to [0,+\infty)$. Let's $B[x_o,b]$ is a compact ball of radius $b$ centered at $x_o$. THEOREM:If ...
0
votes
1answer
14 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
2
votes
0answers
83 views
+50

Consider the equation: $x' = f(t,x)$. Prove that there is a two-way correspondence between the initial and the limits of the solutions.

Consider the equation: $$x' = f(t,x)$$ wherein, $$|f(t,x)| \leq \phi(t)x, \forall(t,x) \in \mathbb{R}\times \mathbb{R} $$ $$ \int^{\infty}_a\phi(t)\,dt< \infty$$ where $a \in \mathbb{R}$. If in ...
0
votes
0answers
14 views

Find the minimizer of the functional

Find the minimizer of the functional $ l= \int u(t) $ with $u(1)=u(1)=0 $ subject to $g=\int $$\sqrt{1+u'(t)} dt $ I want to solve it using E-L equation first $l^*=l- \lambda g$ then i used e-l ...
2
votes
0answers
33 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
7
votes
2answers
239 views
+50

Does Tom catch Jerry?

Tom has Jerry backed against a wall. Tom is distance 1 away (perpendicularly). At time t=0, Jerry runs along the wall. Tom runs directly towards Jerry. Tom always runs directly towards Jerry. Tom and ...
2
votes
0answers
16 views

Equillibria to Differential Equations

I am wondering what the exact definition is of an equilibrium to a differential equation. It seems like the general consensus implies that a differential equation will only have an equilibrium if it ...
3
votes
2answers
528 views

General and particular solution of differential equation

1) I need to find, in implicit form, the general solution of the differential equation $$\frac{dy}{dx}=\frac{2y^4e^{2x}}{3(e^{2x}+7)^2}$$ 2) I then need to find the corresponding particular solution ...
0
votes
0answers
10 views

basic reproduction number of a simple SEIR-model

the normal SEIR-model is: $\begin{array}{rll} \displaystyle{\frac{dS}{dt}}&=\mu N -\mu S -\beta \frac{I}{N} S & \text{Susceptible} \\ \displaystyle{\frac{dE}{dt}}&= \beta \frac{I}{N} S ...
1
vote
2answers
42 views

Repeated root case for ay''+by'+cy=0

To solve the ODE $a y''(t)+b y'(t) +c y(t) = 0$, where $a,b,c$ are constant, we solve the characteristic equation $ar^{2}+br+c=0$. In the case when the roots are two repeated roots, i.e,. ...
0
votes
1answer
19 views

How to solve Sturm-Liouville problem

Find all the eigenvalues and eigenfunctions of Sturm-Lioville problem: $$y'' + (1 + \lambda)y = 0$$ $$y(0) = y \left(\frac{\pi}{2}\right) = 0$$ Can someone please tell me how to solve this? Because ...
0
votes
0answers
27 views

Stability of equilibriums

The question is to find the stability of the equilibriums of the system: $$\frac{dx}{dt}= 8x - 2y - 4x^3 - 2xy^2$$ $$\frac{dy}{dt}= x + 4y - 2y^3 -3x^2y$$ There are 3 equilibriums, $(0,0), (1,1), ...
-1
votes
1answer
172 views

Asymptotic behavior of the solution of $x^4 \frac{d^2y}{dx^2}+ \frac{1}{4}y=0$ near $0$ [closed]

Can you help me find the leading asymptotic behaviors about the irregular singular point $x=0$ of $$x^4 \frac{d^2y}{dx^2}+ \frac{1}{4}y=0$$ So far I have got $y(x) = ...
0
votes
0answers
10 views

Is $x=0$ an ordinary or singular point? Two conflicting textbook solutions that use the same reasoning.

We're asked to determine whether $x=0$ is an ordinary point or singular point for the following two ODEs: $$\begin{align*}x y''+\sin x\,y&=0&(1)\\\\ x y''+(1-\cos ...
0
votes
1answer
47 views

Use series methods to find solution corresponding to..

Use series methods to find solution corresponding to $a_0 = 1$ for the equation $(x+1)y' - y = 0$ Here is my work. Can someone verify that I have the correct solution: So for my final solution I ...
-3
votes
2answers
46 views

Solve $2y^{(5)}-7y^{(4)}+12y'''+8y''=0$ [on hold]

Find the general solution of higher order linear differential equation? Find the general solution of Differential equation using auxiliary equation? $$2y^{(5)}-7y^{(4)}+12y'''+8y''=0$$