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

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2
votes
2answers
58 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 ...
1
vote
0answers
52 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.
0
votes
2answers
51 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) = ...
2
votes
1answer
230 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)
0
votes
0answers
104 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, ...
1
vote
0answers
44 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 ...
0
votes
0answers
36 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
votes
1answer
76 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
vote
1answer
22 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 ?
3
votes
2answers
38 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
votes
1answer
31 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?
0
votes
1answer
192 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
votes
0answers
44 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 ...
0
votes
2answers
55 views

Simultaneous Total Differential Equations 2

To Solve : $\displaystyle \frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz} $ Any hints?
1
vote
0answers
16 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
votes
1answer
40 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 ...
1
vote
2answers
52 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 $(b)$.
0
votes
0answers
48 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 ...
0
votes
1answer
44 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
votes
0answers
72 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?
2
votes
3answers
115 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.
3
votes
2answers
226 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 ?
0
votes
1answer
39 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 ...
5
votes
0answers
62 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 ...
2
votes
3answers
202 views

Solving $r'(x) = \frac{ p(x)-r(x)s'(x) }{ s(x) }$

Can we solve $$r'(x) = \frac{ p(x)-r(x)s'(x) }{ s(x) }$$ with unknown $p(x)$, if we are allowed to pick any $s(x)$ that makes the differential equation easiest? Or, if we need to know $p(x)$, and ...
0
votes
2answers
81 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
0answers
91 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 ...
1
vote
1answer
26 views

Finding the degree and order of differential equations

Find the order and degree of the differential equation $\mid \frac{dy}{dx} \mid + \mid y \mid = 0$
2
votes
2answers
32 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 ...
1
vote
1answer
78 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 ...
3
votes
2answers
41 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: $$ cos \,3y \,dy = -sin \,2x \,dx $$ $$\int cos\,3y\,dy = \int -sin\,2x\,dx$$ $$ ...
1
vote
1answer
53 views

Euler's Numerical Method

Let $\eta(x;h)$ be the approximate solution furnished by Euler's method for the initial-value problem $y'=y, y(0)=1$. I proved that: $i) \eta(x;h)=(1+h)^{x/h}$; $ii) \eta(x;h)$ has the expansion ...
0
votes
1answer
26 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 $ ...
2
votes
3answers
40 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
vote
2answers
46 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 ...
1
vote
1answer
56 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 ...
1
vote
0answers
171 views

Stability of a critical point

I have a question about the stability of a critical point of the system: $$\frac{dx}{dt} = \begin{bmatrix} 4 & -26 \\ 1 & -6 \end{bmatrix}x + \begin{bmatrix} 2 \\ 0 \end{bmatrix}.$$ I have ...
1
vote
0answers
28 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
1
vote
2answers
37 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
0
votes
0answers
29 views

Correct me if I am wrong (homogeneus ODE)

Solve the following ODE: $x y' = \sqrt{x^2 - y^2} +y $ This is my attemp: $y' = \dfrac{\sqrt{x^2 -y^2} +y}{x}$ Now $z = \dfrac{y}{x}$ thus $y' = z'x + z$ and using this change $z' x = \sqrt{1 - ...
1
vote
2answers
80 views

Can we solve $q(x)p'(x)+2p(x)q'(x)=0$ given constraints?

If we suppose that we want $-p'(x)q(x) = f(x)$ for a given $f(x)$, and $$q(x)p'(x)+2p(x)q'(x)=0$$ Can we get $p(x)$ and $q(x)$?
1
vote
1answer
68 views

Partial derivatives-Why does this stand?

In my notes there is the following: $$u_{\xi \eta}=0 \Rightarrow \left\{\begin{matrix} u_{\xi}=0 \Rightarrow u=g(\eta)\\ u_{\eta}=0 \Rightarrow u=f(\xi) \end{matrix}\right.$$ I haven't understood ...
0
votes
1answer
256 views

Snowplow Problem

A snowplow can remove snow at a constant rate (in cubic feet per minute). One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, ...
0
votes
0answers
39 views

Is this divergence-free? (Double Pendulum)

Concerning this page http://scienceworld.wolfram.com/physics/DoublePendulum.html for the double pendulum the moving equations are given by $$ ...
1
vote
1answer
31 views

What is the difference between 1-dim.harmonic oscillator and 2-dim. harmonic oscillator?

I ask myself what exactly is meant with "2-dimensional harmonic oscillator". I only know the situation of a bob hanging on a bar... is that 1-dimensional or 2-dimensional?
1
vote
3answers
421 views

Calculate minimum perimeter of a rectangle with an extra constraint.

I have been set this problem, and although I can derive a minimum perimeter using calculus, I now need to add an extra constraint to one side of the rectangle and I am having problems deriving a ...
1
vote
1answer
44 views

Solving analitically this simple ODE

I could not find with maple an explicit analytical solution (i.e. not involving complicated integrals) to this simple ODE: $\frac{d h(x)}{dx}+C = -\frac{h(x)}{x^2} $ where C is a constant. Is it me ...
3
votes
2answers
76 views

Provide an example of a function whose inverse is also it's derivative.

This is a question from a mathematics competition. I'm totally stumped with this one. If anyone could give an example, or even better, show working, that would be great.
6
votes
3answers
3k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
0
votes
0answers
93 views

Prove a differential equation:

The partial differential equation $$\frac{d^2u}{dt^2}=c^2\left(\frac{d^2u}{dx^2}+\frac{d^2u}{dy^2}+\frac{d^2u}{dz^2}\right) \tag{$*$}$$ is the three dimensional wave equation. In the case of ...