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

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31 views

Transform and numerically solve an ODE with heaviside of form $F'(z) = g(F(z)) + d + c \mathbf{H}(\bar{z} - z)$

I have an ODE in $F(z)$ (really a system of equations, but assume the vastly simplified form here) $$ F'(z) = g(F(z)) + d + c \mathbf{H}(\bar{z} - z) $$ Where $g(\cdot)$ is some non-linear operator ...
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0answers
22 views

Numerical solution to a coupled differentio-algebraic system of equations

$$\frac{\mathrm{d}X_1}{\mathrm{d}t} = P \times ( \frac{I_a^n}{K_i\times exp(I_a*m) + (I_a)^n} ) \times ( 1-( \frac{A.X_2 + B}{ K_o})^z)$$ $$X_1 = X_2 -[ P' \times \frac{I_a^n}{(Ki*exp(I_a * m) + ...
2
votes
1answer
42 views

Finite differences to ODE in polar coordinates

I have an equation of finite differences as follows: $$\frac{X_1(r+\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r+\epsilon} } + \frac{X_1(r-\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r-\epsilon} } = ...
2
votes
1answer
55 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ z z' $?

This is a follow-up to Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?. It turns out that in that case, \begin{align} \text{$ y = ...
2
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0answers
44 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
3
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2answers
149 views

I can't quite figure out this “separable equation”

My prof assigned this question for exam studying and I can't figure it out. It's supposed to be a separable equations question and I'd be able to do something, but for that pesky '$+ y$'. All we've ...
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3answers
56 views

Inverse Laplace Transform,

I have been stuck on this problem for quite a bit, have tried to look at similar answers on website but no help... The original questions is, Solve the IVP $\ y''+y=\sin(t);y(0)=1;y'(0)=0$ I ...
1
vote
1answer
25 views

Non-dimensionalization problem

I am trying to non-dimensionalize this problem but I am getting stuck and would appreciate some guidance. This is the problem: $\displaystyle x^\prime=rx\left(1-\frac{x}{k}\right)-\alpha xy$ ...
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1answer
37 views

Understanding an Equation and how to implement it

A common method for linking language with psychological variables involves counting words belonging to manually-created categories of language. One counts how often words in a given category are used ...
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0answers
50 views

Solution to system of nonlinear ODEs

I have a specific question regarding how the author of this paper obtained the following solution for the system of nonlinear DEs. The system of nonlinear differential equations is: $$ ...
1
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0answers
41 views

Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
3
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1answer
30 views

How to express $z'(t)$ and $w'(t)$ in terms of $z(t)$ and $w(t)$?

I have these functions: $x' (t) = −5x(t) + 2 y(t)$ $y' (t) = 2x(t) − 2y(t)$ where $x(0)=10$ and $y(0)=0$ I am also given these 2 functions: $z(t) = x(t) + 2y(t)$ $w(t) = −2x(t) + y(t)$ First ...
2
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2answers
49 views

How can I get a “better small angle approximation”?

Is there a way to improve on the small angle approximation so that I get a more accurate answer. Is there any way to use a higher order series for the $\sin$ function or to use some sort of series ...
5
votes
1answer
88 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?

Can an arbitrary function $ y: \mathbb{R} \to \mathbb{R} $ always be expressed as $ \dfrac{z'}{z} $ for some differentiable function $ z: \mathbb{R} \to \mathbb{R} $, or are additional conditions on $ ...
1
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0answers
66 views

Solving a system of linear ODEs

Based on my previous post, I have been stuck on this for a few hours now. I want to solve for $x$ and $y$ from the equation $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$ The original two equations ...
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2answers
46 views

Solve the System of Linear Differential equation $\frac{dy}{dt} = Ay$

Consider $A = \begin{pmatrix} 0&1&0\\0&0&1\\0&0&0 \end{pmatrix}$ and y= $\begin{pmatrix} y_1(t)\\y_2(t)\\y_3(t) \end{pmatrix}$ satisfy $\frac{dy}{dt} = Ay$ ; t>0 ; $y(0) = ...
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0answers
43 views

Find the orthogonal trajectories of the family of curves given by $x^2 + y^2 + 2Cy =1$.

Find the orthogonal trajectories of the family of curves given by $$x^2 + y^2 + 2Cy =1.$$ The ordinary differential equation for the family of curves is given by $y'=\frac{2xy}{x^2-y^2-1}$.Therefore, ...
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0answers
27 views

Where can I find the TOC of “Calculus and analytic geometry” by George B. Thomas 4th ed?

I am currently following the course Calculus Revisited, by Prof. Herbert Gross. In his lecture notes he makes references to the book mentioned in the title, by section number, so far I found a copy ...
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0answers
32 views

System of $2$ nonlinear DEs

Please tell me if I'm on the right track for solving the following system: $$\frac{d{U}}{dt}=a - b U -\frac{\beta U V}{U+V} \\ \frac{d{V}}{dt}=\frac{\beta U V}{U+V}-(b+c+d)V $$ Steps: $1.$ I added ...
2
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1answer
66 views

Hint for solving $ y (y')^2 + (x-y) y' - x = 0$

Need to solve the following ODE: $$ y (y')^2 + (x-y) y' - x = 0$$ I don't really know how to start. Any hints?
1
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1answer
19 views

System of differential equations and Cauchy problem

I have this system of differential equations: $$z=y'$$ $$y=-z'-4$$ How would Cauchy's problem look for this equation, if I have z(0)=-4 and y(0)=1 ?
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2answers
36 views

Solve $ 3 e^x \tan{y} \, dx + \dfrac{2-e^x}{\cos^2{y}} \, dy = 0 $ Stupid error somewhere

I am trying to solve the following ODE $$ 3 e^x \tan{y} \, dx + \dfrac{2-e^x}{\cos^2{y}} \, dy = 0 $$ This is my attempt: Its form looks like, $$P(x,y) \, dx + Q(x,y) \, dy = 0$$ so I may be exact ...
3
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1answer
30 views

Solve $y^{2/3}+(y')^{2/3}=1$ other than the direct method?

Is there any way to solve $$y^{2/3}+(y')^{2/3}=1$$ other than just solving for $y'$ and then integrate?
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0answers
15 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
0
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0answers
19 views

Lp estimates from Elliptic Equation

Using the theorem: Let $f \in L^{p}(\Omega)$, $1<p<\infty$, and let $w$ be the Newtonian potential of $f$, $w(x)=\int_{\Omega}\Gamma(x-y)f(y)dy$. Then $w\in W^{2,p}(\Omega), \Delta w=f$ a.e and ...
1
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2answers
51 views

Differential equation with sec

With $(a)$ I got that $-y^2 dx = \sec^2x\ dy$, but it makes no sense. Hence, no Idea how to handle $(b)$.
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1answer
82 views

Calculate the volume of water in glass over time.

For A) I found that volume should be defined by But I got no idea what to do in b) and c)
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58 views

IVT, Runge Kutta system

I'm really getting frustrated with this question. Could someone give me some help to get me started on both a and b?
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1answer
28 views

Wronskian zero implies linear depen

I don't understand the proof of Theorem 4 in these notes: https://people.math.osu.edu/kwa.1/wi11notes/3.3we.pdf Firstly, I don't understand where Abel's theorem is used at all. Also, going through ...
0
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1answer
149 views

The analytical solution for advection-diffusion equation with source term.

We have: $$\frac{\partial w}{\partial t} + a(x) \frac{\partial w}{\partial x} - v \frac{\partial^2 w}{\partial x^2} = f(t)$$ within a domain $x \in [0,1]$ Simplest Sample is $a(x) = 1$ (constant) and ...
0
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1answer
37 views

How can you get eigen- vectors/values/functions in several different contexts?

In differential equations, you can find the eigenvectors/values of a square matrix: A v = \lambda v, where A is a square matrix. You can also find eigen functions of a differential operator, D, in ...
3
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2answers
212 views

On the constant of integration in solving ordinary differential equations

I very much suspect this but I'm not sure if it's correct: In solved differential equations, does the constant 'c' always represent the value of the dependent variable when the independent=0 ?
2
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3answers
102 views

Solve $y' = \frac{1}{x\cos(y) + \sin(2y)}$

I need to solve this ODE $$ y' = \dfrac{1}{x\cos(y) + \sin(2y)}$$ Could you give me any hints? I don't even know how to start.
4
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1answer
89 views

Under which conditions a solution of an ODE is analytic function?

If I'm not wrong there is a theorem that says that if the conditions for Picard's theorem are satisfied, for an ode $\dot x=f(x,t)$, then the solution of the ode is as smooth as $f$. I think I'm not ...
4
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0answers
52 views

Solution of a nonlinear first order ODE

Is it possible to find an analytic solution to the following ODE: $$y\ln(xy)y'+x=0 $$ It is neither separable nor can be made an exact one. I cannot seem to work any substitution either. I've also ...
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0answers
27 views

Hill's problem for moon trajectories.

When we work with the three-body problem, we have a parameter $\mu$ that shows the ratio of the two biggest bodies with $\mu\in(0,1)$. This let's us do practical applications easily. For example we ...
0
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1answer
35 views

Tangent to integral curve

I have an equation like: $$4y'=y(x^2-4x-3)$$ and I have to find the equation to the tangent to the integral curve, which goes through a random point from the square $K =$ $\{-5≤x≤6,-6≤y≤5 \}$. I am ...
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0answers
25 views

Sturm Liouville eigenvalues eigenfunctions

The equation/Sturm Liouville problem is: $$u'' + \lambda u = 0, \quad 0≤x≤\frac{\pi}{2}, \quad u'(0) = 0, \quad u(\frac{\pi}{2}) = 0 $$ I want to find the eigenvalues and eigenfunctions and the ...
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1answer
20 views

Finding the degree and order of differential equations

Find the order and degree of the differential equation $\mid \frac{dy}{dx} \mid + \mid y \mid = 0$
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0answers
45 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
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2answers
25 views

How to solve a homogeneous 2nd order linear DE?

I want to solve this ODE: Given $y=x^2$ is a solution to $x^2y''+2xy'-6y=0$ find the general solution: The answer for the general solution is: $y=Ax^2+B/x^3$ What method do I need to employ to ...
3
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2answers
39 views

Locate my error for this initial value separable differential equation?

The problem is to solve $ sin\,2x\,dx + cos\,3y\,dy = 0, \;\;\;\;y({\pi\over 2}) = {\pi\over 3}$ Here are my steps: $$ cos \,3y \,dy = -sin \,2x \,dx $$ $$\int cos\,3y\,dy = \int -sin\,2x\,dx$$ $$ ...
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1answer
59 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
0
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2answers
68 views

Please solve this differential equation!

$$\frac{dy}{dx}=(2xy-9x^2+(2y+x^2+1)) $$ I am trying to implicitly solve this equation, but I barely know anything about Calculus. So far I have tried to separate into (2xy-9x^2), and solve the ...
0
votes
1answer
21 views

How to solve this initial value separable differential equation?

$$\mathrm y' = {2x\over 1+2y} \;\; , \; y(2) = 0$$ So far, I have $${dy\over dx}(1 + 2y) = 2x$$ $$ 1+ 2y\,dy = 2x\,dx $$ $$ \int 1+2y\,dy = \int 2x\,dx $$ $$ y + y^2 = x^2 + C$$ However, from $ ...
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2answers
40 views

How to solve this ODE (integration factor?)

Im trying to solve the following ODE: $(x+y+1) dx + (2x +2y -1) dy = 0$ In the theory of my book these presented with the form $P(x,y) dx + Q(x,y) dy = 0$ So for my example we have $P(x,y) = x +y ...
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4answers
36 views

Acceleration is given in terms of velocity. Find velocity in terms of time [closed]

Problem is from IB Math HL book from Fabio Cirrito
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1answer
51 views

Solving of the second-order nonlinear differential equation

I'm solving differential equation $2yy''=y^2+y'^2$. I guess it necessary to reduce an order. I try to write equation in terms of $y'=u$. I get the first-order equation, and after i let $u=zy$. But ...
2
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3answers
37 views

How to transform a differential equation to a system of differential equations

Lets say I have a differential equation like $$y''+y+4=0$$ and I have to convert it to a system of first order equations? How is that done. I am interested in the method (and an explanation of it) ...
1
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2answers
33 views

differential equation contain sin(x)

I have a question I would like to know how to work out such differential equation, by hand without using matlab: $x''= A + B \sin(x)$ then $x = ?$ A,B are parameters