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

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2
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3answers
79 views

Solve differential equation $y'''(t)=y(t) y'(t)$.

Solve following diferential equations $$y'''(t)=y(t) y'(t)$$ I would appreciate some help with this problem. Thanks in advance.
0
votes
0answers
27 views

Solve a Green's function problem $L_x{G(x,\bar{x})=\delta(x-\bar{x})}$

I am not sure how to figure this problem out. Solve a Green's function problem $L_x{G(x,\bar{x})=\delta(x-\bar{x})}$ that is bounded as $x\rightarrow0$ and $x\rightarrow\infty$ $L[y] = ...
0
votes
0answers
31 views

How to solve one differential equation with two independent variables in heat transfer.

$$A\frac{ \partial T_a}{\partial t}=B(T_p-T_a)+C(D-T_a)-E\frac{\partial T_a}{\partial x}$$ Where $A, B, C, D, E$ are constants, $t$ is time and $x$ is $x$-axis of the box in which heat transfer is ...
1
vote
1answer
52 views

Use induction to prove that Legendre polynomials solve the corresponding differential equation

I was given a "classical" homework question where I have to prove that the Legendre polynomials solve the differential equation: $\frac{d}{dx}[(1-x^2)\frac{d}{dx}P_n(x)] + n(n+1)P_n(x) = 0$ However, ...
1
vote
1answer
32 views

Series Solution to Differential Equation

Given the series $$1 + \sum_{k = 1}^{\infty} \frac{\beta(\beta - 1) ... (\beta - k + 1)}{k!} x^k$$ how can I find a differential equation for which this series is a solution? I don't have any idea ...
2
votes
1answer
20 views

fixed points for the following system

I'm trying to find the fixed point for the system (see document attached) but it seems so hard and I don't know what Im doing wrong. Can somebody help me with this. I need to find the to look for the ...
1
vote
1answer
41 views

Torricelli's Law with a Sphere (Ignore Mathematica part, already solved)

Edit 1: According to this PDF, second page. The differential equation for a hemisphere becomes $$(r^2h-h^2) h' = -a^2\sqrt{2gh}$$ This differs from the equation I got, which looks like: ...
1
vote
1answer
19 views

Laplace transform using integrals

If I have $g(t)=\int_{0}^t f(\tau) d\tau$. How would I show that $Lg(s)$=$1 \over s$ $Lf(s)$?
0
votes
0answers
18 views

How can I solve thie ODE with boundary condition

How can I solve this: $\frac{d^4G}{dx^4} = \delta(x-\bar{x}) $ Boundary conditions: $G(0) = G'(0) = G''(1) = G'''(1) = 0$ I tried: Let $G(x) = A_1x^3+A_2x^2+A_3x+A_4$ if $ x < \bar{x}$ and ...
-3
votes
0answers
34 views

Solving this second order differential equation.. [duplicate]

$$x^2y''-xy'+y=0$$ How would I go about solving this one?
1
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2answers
115 views

Understanding proof of Peano's existence theorem

I'm studying the proof of Peano's existence theorem on this paper. At page 5 it is said that the problem $$\begin{cases} y(t) = y_0 & \forall t ∈ [t_0, t_0 + c/k] \\ y'(t) = f(t − c/k, y(t − ...
0
votes
1answer
26 views

Undetermined coefficients with tricky particular solution

How could I solve the diff eq $$y''-7y'+4y=2x^3+xcosx+e^x+x^3e^{5x}$$ If undetermined coefficients were to be used, how could I set it up?
1
vote
1answer
37 views

Solving this second order differential equation (Damping mechanism) [closed]

I'm trying to self-teach myself differential equations, but I'm having trouble with second order equations! How can I solve this one? $\frac{3}{32}y''+12y=0$, $y(0)=\frac{-1}{12}$, $y'(0)=2$ ...
0
votes
1answer
21 views

Understand to paragraphs in Chicones book concerning tangent vector fields

I apologize in advance for the length of this question.... Now, consider the sphere $$ S:=\{ (x,y,z):x^2+y^2+z^2=1 \}$$ in $\mathbb{R}^3$. Now in Chicone's book "Ordinary differential equations" on ...
3
votes
1answer
68 views

Find a differential equation for which a series is a solution

This is just a general question. Given a power series in one real variable, is it always possible to find a differential equation for which the series is a solution? If not, when is it possible?
1
vote
1answer
22 views

Solving for Eigenvalues of a Differential Equation

Find the eigenvalues of $$y'' + \lambda y = 0, \; y'(0) = 0, \; y'(1) = 0$$ In my work, I have: For $\lambda > 0$, $y(x) = c_1\cos(\sqrt{\lambda}x) + c_2\sin(\sqrt{\lambda}x)$, $$y'(x) = ...
0
votes
2answers
22 views

Solve $y''-xy=0$ about the ordinary point $x=0$

This is my work so far, please tell me if there are any mistakes: I assumed the solution is of the form $y=\sum_{n=0}^{\infty} x^n$ Then i found the second derivative of the series, substituted the ...
1
vote
1answer
25 views

Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$

Find the eigenvalues of $$y'' + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies ...
0
votes
0answers
28 views

Solutions of fractional linear dynamical systems

The Mittag-Leffler function is defined as: $$ E_\alpha(\tau) = \sum_{k=0}^{\infty}\frac{\tau^k}{\Gamma(\alpha k + 1)}, $$ which can also be defined, analogously, for matrices $A\in\mathbb{R}^{n\times ...
1
vote
0answers
15 views

Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
1
vote
2answers
23 views

$yy'=\sin(t),y(0)=1$ phase portrait

I need to draw a phase portrait for the equation $y(t)y'(t)=\sin(t)$ with the initial condition $y(0)=1$. So far i've found that $y(t)= \sqrt{3-2\cos(t)}$ and ...
-2
votes
1answer
36 views

Solving differential equations [closed]

$$4y'' + 12 y' + 10y = 0,\text{ with $y(0) =-1, y'(0)=5$}$$ $$y'' + 10y' + 25y =0,\text{ with $y(0) = 2, y'(0) = 8$}$$ How do I solve these two? What answer should I get?
0
votes
1answer
24 views

Finding vector valued function solution to differential equation

find the vector valued function x solution to the differential equation system ( X'=AX ) ( X(0) is a 3 x 1 matrix [1;2;3] ) and ( A is a 3 x 3 matrix [2 0 3; 1 2 0; 3 0 2] ) ( X is a 3 x 1 matrix ...
1
vote
1answer
34 views

is there a way to simplify $x^{2} ( v' (x^{n})' )$?

so I have what is probably an algebra question, if I have $x^{2} ( v' (x^{n})' )$ where the ' denotes a derivative, is there a way to simplify this expression?
0
votes
1answer
37 views

Laplace Transformation using Heaviside functions

I'm not very familiar with Heaviside functions so I am struggling with this: I'm supposed to compute $Lu$ where $u''+4u=H(x-0)+H(x-\pi)$ and $H$ is a Heaviside function. Any suggestions are greatly ...
1
vote
0answers
21 views

given a solution of a second order ODE, what is the way to find another linearly independent solution?

So if i'm given a second order linear diff eq. and one of its solutions, what is the way to find another linearly independent solution? Thanks in advance!
1
vote
1answer
13 views

Homogenous diff equations of the form P(x,y)dx + Q(x,y)dy = 0

thank you for taking the time to help me out. I am reviewing a past test while looking in my book to try to figure out how to do this. I understand the way to solve these types of problems, I only get ...
2
votes
0answers
31 views

BVP eigenvalue problem

I am working on the following problem and I am completely stuck: Show that the eigenvalue problem $$ -u''+4\pi^{2}\int_{0}^{1} u(x)\,dx=\lambda\,u $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ has ...
1
vote
0answers
14 views

Laplace transform and differential equations

Given $\frac {d^2y(t)}{dt^2} + a\frac {dy(t)}{dt} = x(t) + by(t)$ Find: a) $ H(s) = \frac{Y(s)}{X(s)}$ b) ROC of the stable function and the correspond h(t) and determine if the stable system is ...
2
votes
0answers
19 views

Stability of an equilibrium solution with 0 denominator

I'm testing the equilibria of a differential equation and found that one has a 0 denominator. Example: $$\frac{dx}{dt}=2x^{(1/2)}-5$$ Which, when you try and evaluate the derivative at 0, you end up ...
2
votes
1answer
48 views

Lyapunov Stability is a problem for me

Let be $ \dot{X} = F(X) $, $F \in C^1( \mathbb{R}^n)$, $P \in \mathbb{R}^n$ isolate singular point. Suppose there exists a family $S_{{r}_i}$ with $ i \in \mathbb{N}$ such that: $S_{{r}_i} = \left ...
0
votes
0answers
104 views

Separation of variables and quantum mechanics

In the book Quantum mechanics by Eugen Merzbacher, third edition, at page 462 he claims that this differential equation (for the unknown operator $F_0=F_0(x,y,z)$) can be solved by separation of ...
3
votes
1answer
60 views

Differential equation with a unknown function $g$ but a known solution.

Say we have an equation $y'+g(x)y=3x, x>0$. We know one solution to this FDE: $y(x) = x^2$ How can we know if there's a solution which satisfies $y(1) = 2$? Questions I'd like answered Q1: ...
3
votes
1answer
42 views

Simple second order differential equation of the form $f''(x)+ h(x) = 0$

The problem is as follows: The movement equation for a pendulum for some mass hanging in a weightless thread with the length $L$ is as follows: $$\frac{d^2a}{dt^2} + \frac{g}{L}\sin(a) = 0$$ Where ...
1
vote
3answers
68 views

solving this 2nd order ODE

Could someone help me with this problem? I have a second order ODE as such: $$\frac{d^2x}{dt^2}+\beta \frac{dx}{dt} = f(t) $$ I am not sure how to solve this linear ODE, was hoping someone could ...
0
votes
1answer
23 views

Behavior of Non-Hyperbolic Equilibria?

So I'm working on a differential equation problem concerning epidemics - we're using the Kermack-McKendrick model. I've reached a point where I need to sketch phase portraits near my equilibria, ...
0
votes
0answers
24 views

Consider the following midpoint method

Consider the following midpoint method $$ y_{n+1} = y_n + hf(x_n + h/2, (y_n+y_{n+1})/2, $$ Use Taylor expansion to show that the local truncation error is $Ch^3$. It is hard for me. And the local ...
2
votes
1answer
57 views

Differential equation $y' = -2x y+ e^{-x^2}\sin(x)$

Can someone help me solve this equation? $$y' = -2xy + e^{-x^2}\sin(x)$$ Progress I have been trying to solve it with the method of integrating factor. I get to the point where: ...
0
votes
0answers
16 views

Proving a function is a solution of a differential equation given one solution.

Suppose $f(t,x) = f(-t,-x)$ for all $(t,x) \in {\mathbb{R}}^2$. Prove that if $u$, defined on $\mathbb{R}$, is a solution of the differential equation $x'=f(t,x)$, then so is the function $v$ defined ...
0
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1answer
23 views

Prove solutions of differential equations are tangent to each other.

Let $u$ and $v$ be solutions of the first-order differential equation $x'=f(t,x)$, both defined on an interval $(a,b)$, and suppose $u(t_o)=v(t_0)$ for some $t_0 \in (a,b)$. Prove that the curves ...
1
vote
3answers
35 views

Differential equation, substitution

By means of the substitute $y = v(x)Y (x)$, where $Y (x)$ is to be specified, solve the differential equation: $$\dfrac{dy}{dx}+\dfrac{y}{x}=\dfrac{y^2}{x}$$ with $y=2$ at $x=1$ Anyone can solve ...
0
votes
0answers
13 views

Local truncation error of a one-step method

I have the following one-step method: $y_{n+1} = y_{n} + \frac{1}{2}h(f(y_{n}) + f(y_{n} + h f(y_{n})))$ I need to calculate the local truncation error: $\sigma_{n+1} = y(t+1) - z_{n+1}$ where ...
1
vote
3answers
35 views

the Differential Equation $y'^2=y$

Is there a solution for the Differential Equation $y'^2=y$ which can be represented using known functions or integrals?
1
vote
1answer
27 views

Estimating convergence with Liapunov function

we have the system: $x' = 2y(z-1) - x^3, y' = -x(z-1) -y^3, z' = - z^3$ and consider its equilibrium at $\vec{o}$. We want to prove that for every non-trivial solution there exists two positive ...
0
votes
2answers
21 views

Find at least one solution in operation notation

Find at least one solution: $$(D^3+1)^5(D+1)^4(D-1)^4y=3e^t$$ In operation notation $(D^3+1)y=\frac{d^3y}{dt^3}+y, (D^3+1)^5$ is this operation done 5 times. Maybe any ideas? I will be grateful.
3
votes
2answers
54 views

Solve the initial value problem: $\frac{dx}{dt}=2t \sin x$; $x(0)=\frac{\pi}{2}$

Solve the initial value problem: $\frac{dx}{dt}=2t \sin x$; $x(0)=\frac{\pi}{2}$ I am almost done. But I cannot seem to solve the equation for x(t) in terms of t. This is what I got: $$ ...
0
votes
1answer
25 views

Finding the general solution to X'=AX with A = [0 1 0; 1 0 0; 1 1 1]?

This was an exam question that I sort of guessed on. Find the general solution to $X' = AX$, where $A = \left[\begin{smallmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & ...
0
votes
1answer
33 views

Can't see how these equations connect.

I am studing for a final and was reviewing an example but can't seem to remember how these equation transform into each other. Can someone explain what steps I am missing Original problem ...
3
votes
2answers
23 views

Use Laplace Transform to solve the following IVP:

I know that this is a somewhat simple problem but I have been having trouble coming up with the little "tricks" that help with Laplace. The problem is: $y''+2y' +5y = e^{-t}\sin(2t)$ where $y(0) = ...
2
votes
1answer
37 views

Integral form of this IVP

How do I show that the following initial value problem $$ xu''+u'+xu=0,\quad u(0)=1,\quad u'(0)=0 $$ has the following integral form: $$ u(x)=1+\int_{0}^{x} t\ln(t/x)u(t)\,dt $$ I am stuck because if ...