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

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3
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1answer
265 views

Solving Shallow water Equations with Hermite polynomials

I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line. The main equation is $$\frac{d^2\eta}{dt^2} + ...
4
votes
3answers
338 views

Theory of the Mathieu Operator

How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ...
2
votes
1answer
122 views

Finding radius of region with Poincaré stability

I am having some difficulty in solving the last part of the following problem: Find the equation of the phase paths of $\dot{x} = 1 + x^2$, $\dot{y} = -2xy$. It is obvious from the phase diagram ...
4
votes
2answers
213 views

Stochastic predator-prey

My system is a simple $P$ vs $I$ foxes- vs rabbits model given by: $$ \begin{cases} \frac{\mathrm{d}I}{\mathrm{d}t}=& \alpha_I+\lambda_IP- \gamma_II -\delta_IPI;\\ ...
4
votes
2answers
401 views

Essay about the art and applications of differential equations?

I teach a high school calculus class. We've worked through the standard derivatives material, and I incorporated a discussion of antiderivatives throughout. I've introduced solving "area under a ...
1
vote
4answers
109 views

Solve $\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$

Solve: $$\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$$ What I have done till now: $$\left(\frac{dy}{dx}\right)^2 -1= \frac{dy}{dx}\left(\frac{y}{x} - \frac{x}{y}\right)$$ ...
1
vote
1answer
134 views

convert a ordinary differential equation to a recurrence relation

How can I create a differnce equation from a differential equation? The step size h is not given and should stay a variable. For example $y' = y^2 + x^2$ with the known value $y(1) = 2$
4
votes
2answers
367 views

Applied ODEs in trajectory problem

I'm having a hard time solving this problem: Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
1
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3answers
3k views

What is a good differential equations textbook?

I have taken a lot of math in university, but chose to omit differential equations. Unfortunately, now I have to read computer science proofs that use them, mostly ODEs, and this is always a struggle. ...
0
votes
2answers
80 views

Laplace of $x^2\frac{d^2y}{dx^2}$

How does one evaluate the Laplace of functions like $t^2\frac{d^2y}{dt^2}$ ? I wanted to solve a differential equation using Laplace Transform resembling: $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y ...
2
votes
1answer
428 views

Solving a second-order equation using Laplace Transforms

I'm trying to solve this second order differential equation using Laplace Transform. The Laplace transform of the equation is as follows: $$I(s) = \frac{E}{s^2+ \frac{R}{L}s + \frac{1}{LC}}$$ I'm ...
0
votes
1answer
372 views

Transitory heat transfer problem with two materials

I need to solve the heat equation in a complicated situation but before I want to refresh (and test if it is still there) my knowledge with a simpler problem: Infinitely long wall composed by two ...
12
votes
1answer
584 views

Recursive solutions to linear ODE.

When finding the solutions to the simple ODE $$ y'- mxy= x^n \text{ ; } y(0) = 0$$ I found the following: Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define ...
1
vote
0answers
119 views

Confusion with periodic orbits, classical dynamical systems

I have a question related to two theorems in the book Differential Equations, Dynamical Systems, and Linear Algebra [Hirsch & Smale,1974]. First let me describe the framework. Let us consider a ...
0
votes
1answer
131 views

determine the interval in which the solution is defined

The ODE is $dr/d(\theta)$ = $r^2/\theta,$ $r(1) = 2$ separable equation and integrate both sides: $\int(dr/r^2$ - $\int d\theta/\theta$ =$0$ -$\frac{1}{r} - ln(\theta) = c$ solve for c: ...
2
votes
2answers
350 views

How to find a solution to a differential equation based on another, given solution

Let's say I have the DE: $$ (x^2 - 2x)y'' - (x^2 - 2)y' + (2x - 2)y = 0 $$ And I have one possible solution to the DE: $$ y_1(x) = e^x $$ How would I go about solving this? I could ...
2
votes
1answer
177 views

Sturm-Liouville systems and the Wronskian

I'm currently reading through Gohberg and Goldberg's treatment of Sturm-Liouville systems in Basic Operator Theory. Define a Sturm-Liouville system to be a differential equation of the form ...
3
votes
1answer
153 views

Why can you determine the stability of a system by taking the eigenvalues of the Jacobian?

Why can you determine the stability of a system by taking the eigenvalues of the Jacobian? I know it's an elementary question but it's been a while. Thank you!
0
votes
1answer
63 views

differential equation nondevelopable

I try to solve this differential equation whose solution seems not to be constructable in power series $y''+(x+a/x^2+b)y=0$, where $a$ and $b$ are some positive real numbers. If one can help ...
2
votes
0answers
123 views

Positive rotational symmetric solution for p-Laplacian

I have the the following problem and I just can't get my head around how to solve it. Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
11
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
2
votes
0answers
233 views

Grand Prix Race

Driver A has boon leading archrival B for a while by a steady 3 miles. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at ta rate proportional to the square of his ...
0
votes
2answers
593 views

Help with a system of linear first-order ODEs using elimination method

$$ y'+3y+4z=2x $$ $$ z'-y-z=x $$ x is independent variable! The solution I get is not the same as the one on Wolfram Alpha ...
1
vote
2answers
212 views

How do we solve such differential equations?

I encountered the following differential equation when I tried to derive the equation of motion of a simple pendulum: $\frac {\mathrm d }{\mathrm d t}(\frac{\mathrm d\theta}{\mathrm ...
2
votes
2answers
224 views

How to solve the ordinary differential equation?

$\displaystyle\frac{d²y}{dx^2}+ \frac{4}{y}\left(\frac{dy}{dx}\right)^2+2=0$ with $y(0) = 1$ and $\displaystyle\frac{dy}{dx} = 0$ for $x = 0$.
2
votes
3answers
139 views

ordinary differential equation: $(f'(z))^2 = c\,f(z)^3 + f(z)^2$

I swear I have seen this type of ODE before, but I can't remember how to attack it. In general, I would like to know how to solve $$\left(f'(z)\right)^m = c\,G(z)^n$$ where $m,\;n \in \mathbb{N}$ ...
3
votes
1answer
703 views

Numerical solving a constrained system of differential equation

I am in trouble on finding a numerical technique to solve the following system of equations $$\ddot q_1(t)=f_1(q_1(t),q_2(t))$$ $$\ddot q_2(t)=f_2(q_1(t),q_2(t))$$ with a constrain of the kind: ...
3
votes
2answers
79 views

How to deduce $g(x)\equiv 0$?

Suppose that a continuous and integrable function $g$,$g\ge 0$ on the interval $[e,f]$ and $A\geq 0$ is a constant such that $$g(x)\le A \int_{e}^{x} g(s) ds,$$ then $g(x)\equiv 0$ for all $x\in ...
2
votes
0answers
178 views

Where did G. W. Hill develop his relative motion equations?

The equations for relative orbital motion are commonly known as "Hill's equations" (also Clohessy-Wiltshire equations), and the citation given to G. W. Hill's 1878 "Researches in Lunar Theory" in the ...
2
votes
2answers
348 views

Question on reduction of order for linear ODEs

If the characteristic equation for a differential equation can be written as $(s-r_1)(s-r_2)$, the substition $z=y'-r_1y$ yields an equation of the form $z'-r_2z=f(x)$. For example, if our ...
4
votes
1answer
451 views

Conditions for a trapping region.

Suppose I have an autonomous system of ordinary differential equations and I want to show that I have a trapping region: a region of phase space which trajectories can enter but can never leave. One ...
2
votes
1answer
105 views

Nonlinear ODE with integral

My question is to find the function $f(t)$ such that $$\frac{df}{dt} = -2f(t)\int_{0}^{t}f(s)\, ds$$ with $f(0) = 1$. My idea is to divide both sides by $-2f$ and differentiate both sides, and then ...
1
vote
2answers
123 views

How do I differentiate this function?

How do I differentiate $y(t)=-[t]+\frac{1}{2}\cdot(1-3^{t-[t]})$, $t \ge 0$, ($[x]$ is the integer part of $x$) in order to verify that it is the solution of the ODE $y' = \log(3) \cdot ...
1
vote
2answers
163 views

Solving $y'' - xy'+(3x-2)y=0$ using power series

I am trying to solve this equation using the series $$\sum_0^\infty a_nx^n$$ $$y'' - xy'+(3x-2)y=0$$ How to do that? I mean that I can replace the variables using the series but then I ...
1
vote
1answer
39 views

DiffEQ with various quadratic terms

I have the following DiffEQ and would like to solve it, $$2F\left( \frac{{d}^{2}}{d{z}^{2}}F\right) -{\left( \frac{d}{dz}F\right) }^{2}+2F\left( \frac{{d}^{2}}{d{y}^{2}}F\right) -{\left( ...
0
votes
1answer
98 views

How can the method of complex exponentials be used to find a complex solution i(t) which varies harmonically in time.

I am WEAK in DE's. I have been trying to understand this question all day. I posted the question as an image since I don't have enough rep points to post it here. ...
0
votes
1answer
188 views

about first order differential equation

Suppose I have a first order differential operator in matrix format:- $$Dx = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \cdots \\ -1 & 0 & ...
1
vote
1answer
123 views

distribution function

Let$(X,\mu)$ be a measure space. $f:X \to \mathbb{R}$ be a measure function. For every $t\in \mathbb{R}$ the distribution function $F$ of $f$ is defined as $ F(t)=\mu\{x \in X:f(x)<t\}.$ I have ...
0
votes
1answer
142 views

Create 'smooth breakpoint function' by using integral?

Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a ...
0
votes
1answer
2k views

stable and unstable point of function

Suppose that we are given following diferential equation $x'=\sin(x)$. Using linear stability analysis, we should find stability points of this equation, we know that ...
1
vote
1answer
64 views

Polynomial of at least degree 3 for this Cauchy Problem

I'm given this : $$y''(x)+\sin(x)y'(x)+x^3y(x)=1+x$$ $$y'(0)=0,\;y(0)=1$$ and I'm asked to calculate a polynomial of at least degree $3$ that approximates the solution of that problem. May I try a ...
3
votes
1answer
176 views

Can all first order ODEs be made exact?

Elementary differential equations classes usually cover exact differential equations. These are equations of the form: $$M(x,y)+N(x,y)y'=0 \qquad \mathrm{such\;that} \qquad \frac{\partial ...
0
votes
1answer
320 views

Strange behaviour with ode45 in matlab, probably some numerical error.

I'm having a problem with a parameter estimation in a non-linear model. I think the culprit is that ode45 (an ode solver in matlab) is not properly solving my ode. It's the in red highlighted part, ...
7
votes
2answers
164 views

Can't seem to solve this differential equation

Disclaimer: This IS homework. So I will outline the steps I've taken an where I'm stuck. I have the following DE: $$ xy' = y + x\cos^2\left(\frac{y}{x}\right) $$ I then rule out the possible ...
0
votes
0answers
322 views

Problem with Newton's Method in solving a System of Equations

I'm trying to use Newton's method to solve the following system of equations, where f and g are functions of x and y. (h,a,f,c,d,b and k are just constants). $f(y,x)=\left[\begin{array}{c} y^{1}\\ ...
1
vote
1answer
76 views

Solve $I[y]=\int_{x_0}^{x_1}y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2} \mathrm dx$ parametrically

If $$I[y]=\int_{x_0}^{x_1}F(x,y,y') \mathrm dx$$ Where $$F=y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2}$$ Then I have shown the Euler-Lagrange equation implies that $$y(1+(y')^2)=2a$$ For some ...
2
votes
1answer
229 views

Find the extremals of $I[y]=\int_0^1(y')^2 \mathrm dt+\{y(1)\}^2$

Could anyone help me find the extremals of $$I[y]=\int_0^1(y')^2 \mathrm dx+\{y(1)\}^2$$ subject to $y(0)=1$ Most crucially I can't work out how to find the boundary $x=1$. I'm trying to go back ...
1
vote
2answers
385 views

How can I solve the differential equation? $y'+y^{2}=f(x)$

$y'+y^{2}=f(x)$ I know how to find endless series solution via endless integral or endless derivatives , and power series solution method if we know $f(x)$. And also I know how to find general ...
1
vote
3answers
207 views

How to separate this differential equation?

I am having trouble separating this differential equation $$xy' + y = x^2 \sqrt{y}$$ I've gotten as far as $$\frac{1}{\sqrt{y}} dy - \sqrt{y} \frac{dx}{x} = x \;dx$$ but I can't finish it. It ...
0
votes
1answer
410 views

Change of variables of PDE

I have a particle of mass $m$ that moves in 2-d in the potential $V(x,y)=\frac{1}{2}m\omega^2(6x^2-2xy+6y^2)$. I have to use the coordinates $u=\frac{x+y}{\sqrt 2}$ and $w=\frac{x-y}{\sqrt2}$ to show ...