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

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0
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1answer
13 views

Is it true that the number of arbitrary constants in the solution always equal to order of the ordinary differential equation?

Is it true that the number of arbitrary constants in the solution (if solutions exist) always equal to order of an ordinary differential equation? If yes, how to "prove" such a statement, if it can be ...
2
votes
1answer
21 views

How to approach this boundary condition? (Laplace equation on annulus)

EDIT: The original problem $$\nabla^2 u =0 \ \ \ \ for \ \ \ 0<a<r<b\ \ \ ,\ \ \ 0<\theta <\frac \pi 2$$ $$u(r,0)=0,\ \ u(r,\frac \pi 2)=f(r),\ \ u(a,\theta)=u(b,\theta)=0$$ My ...
0
votes
1answer
21 views

Equivalence of Dirichlet problems. Gilbarg & Trudinger

I do not understand the proof of theorem 11.4 in the book "Elliptic Partial Differential Equations of Second Order" by Gilbarg & Trudinger. The reason is that I do not understand the text right ...
4
votes
1answer
564 views

How to use the Fredholm alternative in an ODE

I have the following ordinary differential equation $$ \frac{d^2u}{dx^2} + u = \cos x$$ A particular solution to this problem is $x\sin x$, so we can say that $$ u(x) = c_1 \cos x + c_2 \sin x + ...
2
votes
2answers
30 views

Solve the differential equation of brachistochrone

I'm solving the brachistochrone problem. My solution got as far as $y'=\sqrt{k-y\over y}, k={1\over 2gC^2}$. From https://math.berkeley.edu/~strain/170.S13/cov.pdf page 12, I found that the ...
2
votes
1answer
471 views

Comparison of Adams-Bashforth and Runge-Kutta methods of order 4

I have a system of ODE, that must to solve with numerical methods. I solve it by Adams_Bashforth with order4 and Runge-Kutta with order4 methods. Do you know with same length step which methods ...
0
votes
1answer
349 views

Second- and third-order imply first-order with the Euler Method.

Solve the first-order system that satisfies the given initial conditions using the Euler Method for $ y(0.5) $ and $ z(0.5) $, using a mesh size of $ h = 0.1 $: $ y'' - 6 z^{2} z'- y'- x^{3} y = 0; ...
1
vote
1answer
26 views

General Solution of ODE (complex eigenversion)

I am trying to figure out the general solution to the following matrix: $ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ I got a solution, but it is so ...
2
votes
2answers
49 views

How to solve $x^2y'+xy+x^2y^2=4$

I have a problem and I am not able to solve it. I just need a hint what kind of method I should use for this equation. Thank you. $$x^2y'+xy+x^2y^2=4$$
1
vote
1answer
18 views

Confused in regard to Thereom about ordinary point/analytic point

I am having some trouble understand the implication of the theorem $\mathbf{Theorem}:$ If $x_o$ is an ordinary point of the ODE $P(x)y''+Q(x)y'+R(x)y=0$, ( that is $Q/P$ and $R/P$ are analytic at ...
3
votes
1answer
59 views

Differential Equations in Milnor's Topology from the Differential Viewpoint

On page $23$ Milnor states: Let $\varphi$ : $\mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function which satisfies $$\begin{cases} \varphi(x) > 0, & {\rm for}\,\|x\| < 1 \\ ...
1
vote
1answer
82 views

How to find an ODE with prescribed terminal values?

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
0
votes
2answers
19 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
1
vote
0answers
18 views

Differential equations with conditions

Let's say I am given a inhomogeneous differential equation of second grade with 2 conditions. I can receive my solution by adding the solution of the homogeneous part and one solution of the ...
0
votes
1answer
316 views

Solution to $y''-2y'+y=\frac{e^x}{x^2+1}$ using variation of parameters

Find the general solution to this differential equation using the method of variation of parameters. $$y''-2y'+y=\frac{e^x}{x^2+1}$$ I do not understand this problem at all. Can someone please ...
1
vote
2answers
13 views

Differential equation - Graphic solution and limits

You have the following differential equation: $\frac{\text{d}N}{\text{d}t}=0,00029N*(1500-N) \\ N(0)=200$ a) For what $t$ is $N \geq 750$? I have no idea how to solve this differential equation. Is ...
2
votes
2answers
55 views

A property for an ODE

$2\leq n\in\mathbb{N}$. I have no idea how to show that there is a unique solution $y\in C^1([0,T))$ of the ODE \begin{eqnarray} \begin{cases} y'(t)=(1+y(t)^2)\left(1-\dfrac{n-1}{t}y(t)\right)\ \ \ ...
1
vote
0answers
31 views

ODE Separating Variables

When using separating variables to solve $a(x)\beta (y)dx + \alpha (x)b(y)dy = 0$ First suppose $\alpha (x)\beta (y) \ne 0$ everywhere, then it is equivalent to solve $\frac{{a(x)}}{{\alpha (x)}}dx ...
-1
votes
1answer
29 views

solve the inhomogeneous system

solve the inhomogeneous system \begin{equation*} x'=2x+3y-7 \\ y'=-x-2y+5. \end{equation*} How do I find the particular solution? I know the solution to the homogeneous system \begin{equation*} ...
2
votes
1answer
53 views

Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
0
votes
1answer
28 views

How to solve a particular PDE (which reminds of heat equation)

I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where ...
0
votes
1answer
10 views

Uniformly valid solution to boundary layer problem

If there is a boundary layer at $x=0$ and I have found the outer solutions $y^{left}_{out}$ and $y^{right}_{out}$, and the inner solution $y_{in}$. Than how can I put them together to get a uniformly ...
0
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0answers
29 views

How to get SIR epidemic model's differential equation?

I don't know how SIR formulas are calculated of the form: \begin{equation*} \frac{dS}{dt} = -\beta * S * I, \\ \frac{dI}{dt} = \beta * I * S - \gamma * I, \\ \frac{dR}{dt} = \gamma * I. ...
6
votes
1answer
231 views

Bifurcation of integral curves

Consider the following first order ODE: $$\frac{\operatorname{d}\!y}{\operatorname{d}\!x} = x^2 - y^2$$ Despite the fact that this ODE has a very simple expression, it is not solvable in terms of ...
1
vote
1answer
53 views

heat conduction problem

Find the solution of the heat conduction problem $U_{xx} =4U_t , 0 < x < 2, t>0;$ $U(0,t)=0, U(2,t)=0, t>0$; $U(x,0)=2\sin(\frac\pi2x)-\sin(\pi x) + 4\sin(3\pi x), 0 \le x \le 2 $ ok ...
0
votes
1answer
24 views

Question on why this differential equation is solved like this.

This is what it says in my notebook how the following differential equations are solved: $$F(t,x,x',x'',...x^{(n)})=0$$ such that: $$F(t,lx,lx',lx'',...lx^{(n)})=l^kF(t,x,x',x'',...x^{(n)})$$ then the ...
0
votes
1answer
22 views

differential equation and general solution

I have the following differential equation ; $$\frac{\partial z}{\partial t}+\alpha z\left(t\right)=y\left(t\right)$$ I tried to find the general solution by multiplying two sides by $e^{\alpha t}$ ...
0
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3answers
22 views

Can't see a detail within a differential equation. Any help?

Solve: $$xx''=x'^2+x'\sqrt{x^2+x'^2}$$ Answer: $$x'=p(x) \\ x''= \frac{dx'}{dt} = \frac{dp}{dx}\frac{dx}{dt}= p'p\\ \\ xp'p=p^2+p\sqrt{x^2+p^2}\\ p'= \frac{1}{x}p+ \sqrt{1+ (\frac{p}{x})^2} \\ ...
3
votes
0answers
36 views

Maximum principle-estimation

Let $S=\{x \in \mathbb{R}^2 \mid |x| <1\}$. Using the maximum principle I have to show that the solution of the problem $$-\Delta u(x)=f(x), x \in S \\ u(x)=0, x \in \partial{S}$$ satisfies the ...
0
votes
1answer
17 views

Finding the particular solution of a system of differential equation (first order)

From: $$x' -\begin{bmatrix} -7&9 \\ -6&8 \end{bmatrix}x = \begin{bmatrix} 4e^t \\ 3e^t\end{bmatrix} $$ i know that the solution x from this non-homogenous equation consists of a homogenous ...
1
vote
1answer
28 views

Differential equation - can't find mistake

I've got this differential equation: $$xy'=y-x\exp{\frac{y}{x}}$$ I used $\frac{y}{x}=z$ to solve it and the answer I get is $$y=\frac{x}{\ln(\ln(x))}$$ (while it should be $y=-x\ln(\ln(Cx))$. I think ...
1
vote
2answers
38 views

Best methods for solving ODE with series

I am looking for some tips and suggestions in regard to the following problem (post below). I am not sure if I am on the right track, so if anyone could let me know that would be greatly appreaciated. ...
0
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2answers
41 views

solve diferential equation difficulties

I'm studying math and I've founded this equation: $\frac{dp}{dt}=0.5p-450$. I write it so: $p'=0.5p-450$. Derivating the two sides: $p''=0.5p' \Rightarrow p''-0.5p=0$ General solution: $m^2-0.5m=0 ...
-3
votes
2answers
38 views

Differential equation. [on hold]

For my university assignment I need to solve this problem. And I do not know how to do it. $${dx\over dt}={-5x+8\over 6t+9}$$ for $t=6.4$ that satisfies $x(0)=7$.
2
votes
5answers
64 views

The constant of integration in the solution to the differential equation $-4 g(x)=2 x g'(x)$

When I solved this differential equation--- $$-4 g(x)=2 x g'(x)$$ ---I obtained $$\log (g(x))=-2 \log (x).$$ Solving for g(x) I got $\frac{1}{x^2}$. Now this is an error that I constantly ...
0
votes
0answers
45 views

Proof verification+proposition

Given 2 function $F(p,v)$ and $\frac{dF}{dv}=g(p,v)$ Differentiate F(p,v) with respect to v give $F_pf+F_v$ Formula 1 $$\frac{dF}{dv}=F_p\left(\frac{dp}{dv}\right)+F_v=g\\ ...
1
vote
1answer
26 views

Thickness of the Boundary Layer

Given an ODE $$\epsilon y''+2xy'=x \cos(x)$$ with boundary condition $y(\pm {\pi \over 2})=2$ Where is the boundary layer and what is the thickness of it?
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0answers
37 views
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0answers
9 views

Quadruple integral of the solution to a new type of fractional differential equation

Let $\text{D}$ denote the differential operator, and $\text{D}^n$ the $n$th application of $\text{D}$ (i.e. the $n$th derivative) for any positive integer $n$. Note that $\text{D}^0 = ...
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votes
1answer
42 views

Laplace Transforms

Solve the initial value problem for y(t) using Laplace Transforms. $$L\{y''+3y'\}=L\{f(t)\}$$ $$s^2Y-sy(0)-sy'(0)+3(sY-sy(0))=L\{t\}+L\{1\}-L\{u4(t)(t-4)\}-5L{u8(t)}$$ ...
3
votes
1answer
65 views

Differential equation of a pendulum

Consider the nonlinear differential equation of the pendulum $$\frac{d^2\theta}{dt^2}+\sin \theta=0$$ with $\theta(0)=\frac{\pi}3$ and $\theta'(0)=0$. Using the series method, find the first four ...
4
votes
2answers
48 views

How do I solve for $y$ in this differential equation?

$y'(t)= 3ty$ where $y(0)=-1$. I have attempted to solve for $y$ by; $$\frac{1}{y}\space dy=3t\space dt$$ $$\int\frac{1}{y}\space dy=\int3t\space dt$$ $$\implies \ln(y)=\frac{3t^{2}}{2} + c$$ ...
1
vote
3answers
44 views

A method to solve a difficult differential equation

The equation is $\ddot u +4 u =sin^3t $ with the relative cauchy problem $u(0)=1; \dot u(0)=1 $. Solve the homogeneous equation it's not difficult, but my problem is to find out the solutions of the ...
0
votes
1answer
21 views

A problem with a simple PDE

My task is to find a general solution to such a PDE: $xu_x+yu_y=0$. My approach: Such an equation is constant on its characteristics. So at first I want to find out what they look like. ...
0
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1answer
46 views

2nd order linear non homogenous ode

I need to solve a 2nd order linear non homogenous ode which I have no idea how to . This was an example that I neglected to copy down due to not having enough time $$ y"-4y'+13y=6e^{2x}cos{3x} $$ ...
3
votes
2answers
49 views

How to solve this 2nd order ODE

Consider $$\epsilon y''+yy'-y=0$$ with boundary conditions $y(0)=0$ and $y(1)=3$. I showed that the outer solution is $y_{in}(x)=x+2+O(\epsilon)$. Than for the inner solution, I wish to solve the ...
0
votes
2answers
43 views

Integrate $ \int^{\pi}_{-\pi} (\pi^2-x^2)\sin nx \ dx$

Consider the function $f:(-\pi,\pi)\to\mathbb{R}$ be defined as $x \mapsto (\pi+x)(\pi-x)$ Compute the fourier series of $f$ So far, I've worked out $a_o$ by: \begin{equation} a_o = \frac{1}{\pi} ...
0
votes
2answers
25 views

Convert from complex exponentials to sinusoids

I'm working through some notes on signals and systems, and got stuck trying to fill in the missing steps in converting the left hand side to the right hand side of the following equality: $$ \alpha_i ...
0
votes
0answers
12 views

Lyapunov function for discrete dynamical system

Consider the ODE \begin{equation} \dot{x}(t) = h(x(t)), \end{equation} where $h: \mathbb{R}^d \to \mathbb{R}^d$ is a continuously differentiable map. Let $x^*$ be an asymptotically stable ...
0
votes
0answers
17 views

Numerical Methods: Mid Point Higher Order ODEs

I am taking a Numerical Methods class and the professor told us to find out how to solve Higher Order Ordinary Differential Equations using the midpoint method. As of right now, I only know how to use ...