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

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1answer
55 views

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ [on hold]

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ Consider a 1st order differential equation $dy/dt + 409y = f(t)$ a)Solve and write the general ...
2
votes
1answer
43 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
1
vote
1answer
14 views

Differential Equations: Stable, Semi-Stable, and Unstable

I am trying to identify the stable, unstable, and semistable critical points for the following differential equation: $\dfrac{dy}{dt} = 4y^2 (4 - y^2)$. If I understand the definition of stable and ...
3
votes
2answers
52 views

How to determine generalized eigenvectors of $\begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 &1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$

I want to calculate the general solution of this DE-system: $$ \frac{d \vec x}{d t}= A \vec x,\text{ with }A = \begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 ...
0
votes
1answer
14 views

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? [on hold]

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? Any help or just tips will be appreciated, thanks.
1
vote
1answer
37 views

how to show that all solutions tend to zero?

Here is our nonlinear first order ode: \begin{equation*} y'(t) +2y(t)+y^3(t)=e^{-t} . \end{equation*} We want to show that all solutions tend to zero as $t$ goes to infinity. Attempt: Multiply both ...
0
votes
1answer
14 views

System differential equations 0

System of nonlinear differential equations $$y'= -\frac{4y}{x+4}+\frac{y^2x}{4t}, $$ $$ x'= \frac{x^2}{t^2}-\frac{9x}{t}+24 $$ help if I understood correctly you need to express $x$, but I can't
1
vote
2answers
40 views

Sane solution for an ODE with physical interpretation

I have an object which is being subjected to a continual force that is a quadratic function of the object's velocity, ie, $F=f_0+f_1 v + f_2 v^2$ for arbitrary but given constants $f_0$, $f_1$, and ...
0
votes
0answers
10 views

Question about phase shift on multiple-scale analysis

Consider the following ODE $$y''(t) + y(t) + \epsilon y^2(t) y'(t) = 0$$ for $t>0$ with boundary condition $y(0)=1$ and $y'(0)=0$ I have found the leading order asymptotic expansion, that is ...
0
votes
0answers
22 views

first-order nonlinear ordinary differential equation0

How to solve this differential equation: $$(x^{2}+\ln(y))\cos(2x)+\sin(2x)(xdx+\frac{dy}{2y})=0 $$ I tried to rearrange the equation to the form $\frac{dy}{dx}$ but I couldn't.
0
votes
1answer
12 views

Learning spectral methods in numerical analysis

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 ...
0
votes
0answers
26 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
votes
3answers
32 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 ...
1
vote
1answer
23 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 ...
0
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0answers
9 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.
0
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3answers
44 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} = ...
0
votes
1answer
18 views

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

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 ...
0
votes
0answers
33 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 = ...
0
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1answer
18 views

Bessel Functions of Half-Integer Order

I recently came across the general form of Bessel Functions of half-integer order given by: $$ ...
2
votes
2answers
23 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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votes
0answers
17 views

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
2
votes
1answer
34 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
votes
1answer
53 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
votes
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
votes
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
votes
1answer
36 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; $$(x^2-36)y''+(6-x)y'+(x^2+12x+36)y=0$$ ...
3
votes
1answer
38 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
votes
1answer
30 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 ...
1
vote
1answer
15 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 ...
0
votes
2answers
35 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
votes
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
votes
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!
1
vote
1answer
23 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
votes
1answer
28 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
votes
2answers
42 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
votes
3answers
52 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
54 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
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
22 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
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 ...
1
vote
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 ...
1
vote
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) ...
0
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0answers
41 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
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 ...
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
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
votes
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} \\ ...
0
votes
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
60 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)\ \ \ ...