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

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2answers
34 views

How can I solve $y$ in differential equation?

$xy'(x)=y(x)(x+1)$ where $y(1)=2e$ I've no idea whatsoever to begin and get an answer! Hints are welcome!
0
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1answer
16 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 ...
0
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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 ...
2
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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 ...
2
votes
3answers
38 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
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
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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 ...
1
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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 ...
1
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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 ...
1
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0answers
33 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 ...
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0answers
85 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) ...
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0answers
31 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. ...
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
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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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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} \\ ...
0
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1answer
31 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 ...
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
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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*} ...
1
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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 ...
0
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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 ...
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 ...
1
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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. ...
1
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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 = ...
3
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1answer
40 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 ...
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votes
1answer
43 views

Laplace Transforms

Solve the initial value problem for y(t) using Laplace Transforms. $$L\{y''+3y'\}=L\{f(t)\}$$ $$s^2Y-sy(0)-sy'(0)+3(sY-sy(0))=L\{t\}+L\{1\}-L\{u4(t)(t-4)\}-5L{u8(t)}$$ ...
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
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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. ...
4
votes
2answers
49 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
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3answers
45 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 ...
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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 $x(0)=7$.
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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
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
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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 $$ y"-4y'+13y=6e^{2x}cos{3x} $$ ...
0
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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 ...
1
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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 ...
1
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0answers
10 views

Nonlinear Schrodinger equation - modified

I have a nonlinear Schrodinger equation: $ia_1\dfrac{\partial A}{\partial x}-a_2\dfrac{\partial^2 A}{\partial t^2}+|A|^2A=0$, $A$ is the amplitude and the above equation governs the slow modulation of ...
1
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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
24 views

$(1-x^2) y^{\prime \prime}-2xy^{\prime}+12y=0$ [on hold]

(1) Find all singular points and classify whether irregular or regular. (2) Find a power series solution about point $x=0$ (3) Write out first 4 terms of solution that have non-zero coefficients ...
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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
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2answers
16 views

Nonhomogeneous Euler-Cauchy equation $x^2 y''-2x y'+2y=4\log x$ [on hold]

Determine solution of the associated homogeneous equation Use method of variation of parameters to find particular solution of non-homogeneous equation. State the general solution of the ...
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0answers
12 views

what is meant by “value of approximate numerical methods or method of digital computer” for finding solution of differential equation?

I was reading "motion against resistive forces" in Newtonian Mechanics: M.I.T. introductory physics series by A.P. French; here is the excerpt: [...] In general, the resistive force $\mathbf{R}$ ...
2
votes
1answer
41 views

Problems with a Simple Differential Equation

I am trying to solve the following: $y' = (y-5)(y+5)$ if $y(4) = 0$. So far, I have tried separating the variables and then use partial fractions and have followed these steps: (1) $A(y+5) + B (y-5) ...
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0answers
31 views

Laplace transform: $y"+6y'+25y=100\sin(10000t)$ [on hold]

Can't figure out how to do the following problem: Find $y(t)$ of $y"+6y'+25y=100\sin(10000t)$ $y(0)=5$, $y'(0)=10$
3
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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 \\ ...
2
votes
4answers
73 views

What does d(something) mean?

In a book I am reading on differential equations, the author writes the following: $$e^{\int P(x) \mathrm{d}x}\mathrm{d}y+P(x)e^{\int P(x) \mathrm{d}x}y\mathrm{d}x=Q(x)e^{\int P(x) ...
1
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1answer
30 views

eigenfunction and eigenvalue

how can I find the eigenfunction for this BVP? $$y'' + (\lambda)\, y = 0, \,\, y(0)=0, y(2\pi)=0$$ with this case $\lambda >0$. I got the general solution: $$y(x)=c_1\cos(\sqrt{\lambda}x) + ...
2
votes
1answer
42 views

General solution of $(x^2-y^2)dx + (3xy)dy = 0$

Find the general solution to the homogeneous differential equation $$(x^2-y^2)dx + (3xy)dy = 0$$ The differential equation does not seem to be separable, and I'm having a tough time to put it in the ...
1
vote
1answer
38 views

Relatively simple system of nonlinear ODEs

There are a lot of questions like this on MSE as well as online resources on the subject, but a) the MSE questions are either unanswered or correspond to systems substantially different from this one, ...
0
votes
1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...