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

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1answer
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Numerical Analysis: Spectral Methods

I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not ...
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0answers
22 views

limit problem-equation

H, I have this problem $$c^2 U''(x)=F(x),\quad U(0)=A,\quad U(\ell)=B$$ $F$ is done, and $0 < x < \ell$ I read that we must found that $$U(x) = A + (B-A)\frac{x}{\ell} + \dfrac{x}{\ell} ...
0
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3answers
27 views

Particular solution of y'' -3y' + 2y = e^t

I'm trying to find a particular solution of $$y'' -3y' + 2y = e^t$$ My fundamental set is: $$y_1 = e^{2t}\\y_2 = e^t$$ So I chose $y_p = A t e^t$, which gives me:$$y_p' = Ae^t + Ate^t\\y_p'' = 2Ae^t ...
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0answers
7 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
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0answers
7 views

Do you know how get differential equations of HSIR model of propagation malware? [on hold]

I have differential equations but I don't know how get it? thank you for help me.
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3answers
42 views

Solve the following second-order differential equation: $\ddot{x} + \dot{x} = 5t\cos(t) + 4\sin(t)$

I am trying to solve the following second-order differential equation: $$\ddot{x} + \dot{x} = 5t\cos(t) + 4\sin(t). (*)$$ I know that if the equation had instead been: $$\ddot{x} + \dot{x} = ...
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1answer
15 views

How to find solution to $y'=y_1(x)g(x)+y_2(x)f(x)$?

Asuume that function $y=y_1(x)$ is one of the solutions of differential equation $y'=f(x)$ as well as $y=y_2(x)$ of $y'=g(x)$. You need to find at least one solution of this equation: ...
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0answers
29 views

Confusion regarding dF/dx=0, F=constant

I thought i found a theorem "Given a curve in the (y,x) plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of F along this curve satisfies relation $g = ...
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1answer
15 views

Bessel Functions of Half-Integer Order

I recently came across the general form of Bessel Functions of half-integer order given by: $$ ...
2
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2answers
18 views

Need help with linear ODE, indicial and recurrence.

I am having trouble understanding something and I want to post what I have done so hopefully someone can catch where I have made a mistake. The question asks; determine the indicial equation, ...
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0answers
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Let v take any arbitrary value define y extraneously [on hold]

Verify: f'(x) O f'(y) e^x ln 2x = bc Tan (2x)^v(4C2). Quite easy but lengthy. here, Let v take any arbitrary value define y extraneously
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1answer
33 views

Equilibrium solutions for $y'=t^{3}y$

I'm having trouble understanding the following. To solve the differential equation $$y'=t^{3}y$$ I go about it in the following way: \begin{align*} y'&=t^{3}y\\ \frac{y'}{y}&=t^{3}\\ ...
2
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1answer
52 views

Confusion about ODE

so I am in a class for ODE and for me is is moving a bit quick. I am one year behind most of the class but thats note anything rare. But I am feeling very stumped on something now. Because, usually I ...
2
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0answers
33 views

Continuation of differential equation

Suppose I have a differential equation $$\dot{x} = f(x)$$ which has global solution for any initial value $x(0) \in \mathcal{S}$. Is there some theorem defining conditions under which this equation ...
0
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3answers
32 views

Determine the form of solution to differential equation, for particular starting value

I am working on a differential equations problem. I must first find the general solution to: $$y' = y(y-1),$$ where $x$ is the dependent variable. I have managed to solve this, to get the answer: ...
3
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1answer
35 views

Behaviour of solutions to ODE near singular points

I am having trouble understand how to classify what happens to solutions of ODE near singular points. For example, I have a question that is about the ODE; $$(x^2-36)y''+(6-x)y'+(x^2+12x+36)y=0$$ ...
2
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1answer
35 views

How do I go about solving this differential equation?

$$t^2x''-(6t^4+2t)x'+9t^6x=0$$ I was taught to write as the following $x= t^n+a_1t^{n-1}... \\ x'=nt^{n-1}+a_1(n-1)t^{n-2}...\\ x''=n(n-1)t^{n-2}+a_1(n-1)(n-2)t^{n-3}...$ And then plug those into ...
0
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1answer
28 views

Laplace diffrential equation

$$\frac{dx}{dt}=2x +3y$$ $$\frac{dy}{dt}=3x +2y$$ Find general solution. I know there is a solution through eigenvalues. But I want to solve it with Laplace transformation. I almost get the right ...
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1answer
14 views

Question about Frobenius Method

I am having some confusion and looking for some help/suggestions about the following. Consider the ODE; with regular singular point $x_0=0$ $$2x(x-1)y''+3(x-1)y'-y=0$$ And I am supposed to find the ...
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2answers
30 views

Solving a higer order differential equation

Let $n=1,2,3\dots$ Discuss how the observations $D^n(x^{n-1})=0$ and $D^n(x^n)=n!$ can be used to find the general solutions of the given differential equations. $y''=0$ $y'''=0$ $y(4)=0$ $y''=2$ ...
2
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2answers
27 views

Solve of the differential equation $y'=-\frac{x}{y}+\frac{y}{x}+1$

I've tried to solve this equation, and in the course of solving any problems. Please help me understand. $$y'=-\frac{x}{y}+\frac{y}{x}+1$$ Results in a normal form. ...
2
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2answers
29 views

Solve of the differential equation $\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$

I have some problem. There is an equation: $$\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$$ Open brackets. $$3y^2dy+x^2dy+2xdx y+xdy +ydx +2ydy+dy=0$$ But what to do, tell me, please? I saw this a ...
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2answers
45 views

How can I solve $y$ in differential equation? [on hold]

$xy'(x)=y(x)(x+1)$ where $y(1)=2e$ I've no idea whatsoever to begin and get an answer! Hints are welcome!
0
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1answer
22 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 the order of an ordinary differential equation? If yes, how to "prove" such a statement, if it ...
0
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1answer
26 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 ...
3
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1answer
36 views

How to address multiple cases in this BVP? (Laplace equation in quarter-annulus)

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
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3answers
50 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 ...
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2answers
53 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
19 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 ...
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0answers
21 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 ...
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2answers
16 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 ...
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0answers
42 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
88 views

How to find an ODE with prescribed terminal values? [on hold]

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
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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
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1answer
26 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 ...
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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 $e^{\alpha t}$ ...
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3answers
25 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} \\ ...
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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
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2answers
59 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)\ \ \ ...
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1answer
31 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
29 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
18 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 ...
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2answers
42 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 ...
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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. ...
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0answers
11 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
44 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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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
65 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
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. ...