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

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0
votes
3answers
94 views

Difficult first order ODE: $x't(x'+2)=x$

$$x't(x'+2)=x$$ I tried to transform it into homogenous equation, but that doesnt seem like a right approach. Any help would be appreciated.
2
votes
0answers
38 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
5
votes
2answers
82 views

Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
1
vote
2answers
75 views

Hard ODE: $x^2(y')^2+3xyy'+2y^2=0$ [closed]

$$x^2(y')^2+3xyy'+2y^2=0$$ I have no idea how to start, I probably need to do some tricky substitution but as of now I cant see any options.
0
votes
0answers
47 views

Finding the largest eigenvalue of a sparse matrix

I would like to find the largest eigenvalue of a sparse matrix by hand- this is part of analyzing a mathematical model for infectious diseases. The nonzero entries are very complicated - hence Maple ...
1
vote
3answers
212 views

Solve the differential equation: $x\dfrac{dy}{dx}=\dfrac{1}{y^3}$

Would anybody be able to help me work through this problem? I am having some trouble with it. $$x\dfrac{dy}{dx}=\dfrac{1}{y^3}$$
1
vote
1answer
45 views

A problem with non-exact ODE

$$xy^3dx=(x^2y+2)dy$$ After switching everything to one side $$xy^3dx-(x^2y+2)dy=0$$ $$\frac{\partial P}{\partial y}=3xy^2$$$$\frac{\partial Q}{\partial x}=-2xy$$ $\frac1P(\frac{\partial Q}{\partial ...
2
votes
2answers
137 views

Checking some work on differential $y''-4y=\sinh x$

I want to check that I have done this correctly, because I feel like I am most of he way there but missing something important (or maybe just something obvious that I ought to know) We have the ...
1
vote
1answer
32 views

How to show that the one step method can't have consistency $p=3$?

I was looking at some exercises from last years' of my Intro to numerical math class, and found this: Consider the following explicit one step method: $$\psi^h x=x+h \gamma_1 f(x)+ h \gamma_2 ...
1
vote
2answers
42 views

Finding family of curves

Find family of curves for which lenght of the line segment between origin and the point in which line normal to the curve intersects with X axis is equal to $\frac{y^2}x$ The equation should be ...
2
votes
6answers
124 views

Differential equation $2xy \frac{dy}{dx}-y^2+x=0$

I've been doing some exam tasks and I've come along this equation $$2xy \frac {dy}{dx}-y^2+x=0$$ that I dont know how to solve. I probably need some substitution in here but I just can't see it.
-1
votes
1answer
45 views

Help with first order ODE

Find particular solution of equation $$x'=x^2-tx-t$$ knowing that its general solution is in form $x(t)=ct+d$ First off I evaluated $c$ and $d$ to be $1$ and $1$ by substituting given solution to the ...
3
votes
2answers
130 views

second degree differential equation

Find all functions $f(x)$ such that $$f''(x)+f(x)=\frac{1}{1+x^2}.$$ I would like to know if it's solvable and the solution/hints. What I got : ...
0
votes
2answers
70 views

Solving coupled first-order linear ODEs

Basically this question comes from population modelling. Let y be the population of Lions and let x be the population of deer. By ignoring the effect of deer, we observe that $$dy/dt = k_2 y$$ ...
1
vote
1answer
75 views

Delayed System Help

It is well-known that a small delay may or may not cause stable equilibrium to become unstable. Can anyone help that if for $\tau=0$ the equilibrium solution is unstable and if $\tau>0$ is there a ...
3
votes
1answer
58 views

Every equation of the form $ax''+b(x^2-1)x'+cx=0$, $a,b,c>0$ can be transformed into Van der Pol's equation

Show that every equation of the form $$ax''+b(x^2-1)x'+cx=0,$$ $a,b,c>0$ can be transformed into Van der Pol's equation by a change in the independent variable. I am unable to find this ...
0
votes
1answer
50 views

Integral with square root of function of function

I have the function $y=y(x)$ with $y'=dy/dx$, and the following equation: $ky'=\pm\sqrt{k^{2}-y^{2}}$, where $k$ is constant. Integrating this, given that $y(0)=0$, should give: $y=k\sin(x/k)$. I ...
0
votes
1answer
159 views

Book for differential equations

I generally use Rudin's book to prepare for my analysis lectures, however, we started doing Lagrange multipliers and differential equations (e.g. Picard-Lindelöf Theorem) which unfortunately isn't ...
1
vote
0answers
77 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
1
vote
1answer
139 views

Finding functions in a differential equation that satisfies two solutions

"Given a differential equation of the form $a(x)y''+b(x)y'+y=0$, find functions $a(x)$ and $b(x)$ so that $y=x$ and $y=x^2$ are each a solution of this differential equation." I'm really not sure how ...
0
votes
1answer
224 views

Find the solution of the following second order nonlinear ODE.

I was asked by my friend to solve an ODE which is reduced to the following form: $\frac{d^2 y}{d t^2} exp(2y) + f(t) =0 $ where $f(t)$ is a given positive function. But I cannot go further. Would you ...
2
votes
0answers
124 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
6
votes
3answers
562 views

Why do we chose exponential function as a trial solution for second order linear differential equation with constant coefficient?

Why do we chose exponential function as a trial solution for second order linear differential equation with constant coefficient ? Can any other function be taken as a trial solution ?
1
vote
0answers
43 views

About the maximal interval of existence

Let $f:\mathbb R\times \mathbb R^n\longrightarrow\mathbb R^n$ be a continuous function such that there exists some $T\in\mathbb R$ with the following property: $$f(T+t,x)= f(t,x)\;\;\forall ...
2
votes
2answers
73 views

How to determine the eigenvectors for this matrix

I have the matrix $$\left( \begin{array}{ccc} -\alpha & \beta \\ \beta/K & -\alpha/K \end{array} \right)$$ for which the eigenvalues are ...
1
vote
2answers
54 views

On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...
2
votes
2answers
42 views

Find $p(t)$ if $p'(t) = Ap(t) + Bp(0)$

Given $$ p(0) = 800000, $$ $$ p'(t) = Ap(t) + Bp(0), $$ where $A, B$ and $p(0)$ are constant and $p'(t)$ is the derivative of $p(t)$ at $t$. I'm trying to work-out what $p(t)$ is. I know that if ...
0
votes
1answer
48 views

Find eigenvalues and eigenfunctions of this BVP: transforming the eqn?

so I've been stuck on this problem after my first attempt. I got the trivial solution after using the characteristic equation $3r^2 - 4r + 1 = 0$... Find all eigenvalues and associated eigenfunctions ...
3
votes
3answers
74 views

Understanding the Euler operator

While reading this book I came across a differential equation $$t^5\frac{d^2y}{dt^2}+2t^4\frac{dy}{dt}-y=0$$ that was then rewritten in terms of the Euler operator, $\delta=t\frac{d}{dt}$, with the ...
1
vote
1answer
156 views

linearize a nonlinear ode

Could anyone suggest me how to linearize the following system of nonlinear odes (special attention to (2) \begin{align} -cU'&=-U''+UV\tag{1}\\ -cV'&=-k(k+1)V^{k-1}(V')^2+(k+1)V^k ...
1
vote
2answers
44 views

Is this system of differential equations linear?

Suppose I have a system of differential equations like below : $\dot{x} = x + y + 5$ $\dot{y} = x - y $ Is this system linear or nonlinear differential equations?
1
vote
1answer
158 views

How to prove the asymptotic stability of the trivial equilibrium of this system?

I was trying to prove the asymptotic stability of the trivial equilibrium $(0,0)$ of the two-dimensional non linear ODE system: \begin{align} \frac{dH}{dt}=\mu\frac{(H+F)^2}{K^2+(H+F)^2}-d_1 ...
1
vote
1answer
71 views

A phase diagram outlining

I'm trying to solve this differential equation $$x^{ \prime}=f(x)-nx-y$$ $$y^{\prime}=\frac{(f^{\prime}(x)-r)y}{\alpha}$$ where $f:[0,+\infty[\rightarrow \mathbb{R}_{+}$ is an increasing and concave ...
2
votes
1answer
37 views

The number of independent solution of a differential equation

Give a differential equation of order $n$ which has infinite independent solutions. Is it possible? In other words, Is the solutions space dimension of a differential equation always smaller than its ...
1
vote
1answer
311 views

Existence and uniqueness for a system of first-order PDE

Let $y$ be a scalar and ${\bf t}=(t_1,\ldots,t_K)$ and \begin{align*} \frac{\partial y({\bf t})}{\partial t_k} &=f_k(y({\bf t}),{\bf t}) \qquad k=1,\ldots,K\\ y(t_{10},\ldots,t_{K0}) &=y_0 ...
2
votes
1answer
26 views

Linear system with special condition

Consider the linear system $$ x ^ {\prime} = \begin{pmatrix} 3 & \sqrt{2}\\ \sqrt{2}& -2\\ \end{pmatrix}\circ x $$ Does the system has solution for any initial value $ x (0) = \displaystyle ...
3
votes
2answers
60 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
3
votes
2answers
230 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
1
vote
1answer
70 views

Which numerical method to use for ODE?

In practice what is the most common way to numerically estimate $y(t)$ (possibly using a series expansion) in the ODE with initial conditions, $$ y'(t) = f(t,y(t)), \qquad y(t_0)=y_0 $$ Wikipedia has ...
0
votes
1answer
60 views

Laplace transform of a differential equation

Given the Laplace transform \begin{align} \mathcal{L}\{g(r)\} = f(t) = \int_{0}^{\infty} e^{-tr} g(r) \ dr \end{align} can it be shown that the transform of the differential equation \begin{align} ...
1
vote
1answer
40 views

Alternate solution to differential equation

Here is my system of DEs. $$ \begin{align} & \frac {dx} {dt}=2(x-y) \\ & \frac {dy} {dt}= y-x \end{align} $$ Upon solving this, I get the following: $$X=C_1+C_2 \begin{pmatrix}-2 \\1 ...
0
votes
2answers
57 views

Why would I want to find the rate at which things were changing? Marginal cost, marginal revenue and profit

I'm learning calc and after learning about how to differentiate using product rule and chain rule etc. I came across marginal cost and marginal revenue. I'm pretty familiar with cost, profit and ...
1
vote
4answers
112 views

Solve $y = 2 + \int^x_2 [t - ty(t) \,\, dt]$

While working on some differential equation problems, I got one of the following problems: $$y = 2 + \int^x_2 [t - ty(t) \,\, dt]$$ I have no idea what an integral equation is however, the hint ...
2
votes
3answers
46 views

Linearly Independency of functions

Show if the functions are linearly independent $x(t)=3$, $y(t)=3\sin^2t$, $z(t)= 4\cos^2t$ How can i show this?
0
votes
0answers
45 views

Minimizing a multivariable function in several variables

I would like to show that a certain function is negative, to help establish asymptotic stability via a Lyapunov function for a system of differential equations. This is exactly what I need help on: ...
1
vote
3answers
79 views

Solving an ODE using Laplace Transforms

$$y′′′′ + 2y′′ + y = \sin x$$ $$y(0) = y′(0) = y′′(0) = y′′′(0)= 0$$ After solving I got $y(s)=\dfrac1{(s^2 + 1)^3}$ for which I am unable to find the inverse Laplace transform. Please let me know ...
3
votes
4answers
671 views

How can I solve this O.D.E.?

I need to find the solution of $$x''-2x'+x=\sum_{n=1}^Ne^{-nt}$$ I was thinking undetermined coefficients. Is there another way?
1
vote
1answer
58 views

Solution to a second order non-linear ODE

I need help solving two (unrelated) non-linear second order ODE's. Let me just preface this by saying that I don't know much/anything about ODE's (I'm only 17). Well, so here's the first equation: ...
1
vote
2answers
86 views

Units of time in simple differential equation

Very simple question: There is a well-known model in epidemiology called SIR model. It describes the changes in the number of susceptible, infectious and recovered individuals in a population. It is ...
1
vote
2answers
235 views

Eigenvalue and Eigenfunction for a boundary value problem

I'm having trouble understanding some of the concepts related to these problems. Here's an example I'm working on: $$y''+(\lambda+1)y=0 ; y'(0)=0,y'(1)=0$$ The characteristic equation I found was ...