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

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what happens when expansion parameter is of the order of dynamical variable itself?

Lets consider following differential equation, $\epsilon \frac{dy}{dt} = ....$ In principle one can use Method of matched asymptotic expansion or Method of multiple scales to solve such singular ...
7
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2answers
57 views

Can $\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval containing $0$

If $p(x)$ and $q(x)$ are continuous functions for any $x$, can $y(x)=\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval $I=[a,b] $containing $0$? I think it is not as ...
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0answers
22 views

Trying to model a substance settling in water using an advection equation?

I am trying to model a substance dispersed in a container of water gradually settling at the bottom. I am considering only one dimension. The top is at $z = 1$, and the bottom is at $z = 0$. So at $t ...
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0answers
13 views

Perturbative expansion of eigenvalues

Consider the differential operator given by $L_{\epsilon}u := -u'' + \epsilon xu$ with $u(0) = u(\pi) = 0$. For $\epsilon = 0$, then the smallest eigenvalue of $L_0$ is $1$ with eigenfunction $\sin ...
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1answer
18 views

Is this correct solution of higher order differential equation?

Solve the equation $$\require{cancel} (D^4 + 4)y = 0.$$ Solution: The auxiliary equation is: $$D^4 + 1 = 0.$$ $$D^4 = -4.$$ $$(D^{\cancel{4}})^{\cancel{\frac{1}{4}}} = (-4)^{\frac{1}{4}}$$ $$D = ...
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1answer
23 views

Solving a pair of ODEs

I'm trying to solve a pair of ODEs for which I've obtained a solution. However, my problem is that my answer is slightly different from mathematica's answer. $$ \frac{dA}{dt} = \theta - (\mu + ...
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21 views

Reduction of ODE to Bessel [closed]

Please solve the following question Q. Reduce the following equation to the Bessel's differential equation y′′+x²y=0
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27 views

Is the given system of differential equations solvable?

I am trying to implement a system of differential equations (equations of motion of a roll axis vehicle, which are part of a vehicle model) in Matlab but it does not work for me. I have figured out ...
7
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2answers
197 views

What is the value of $x$ such that $\frac{\text{d}^2y}{\text{d}x^2}=0$ where $\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$?

How can you find the values of $x$ such that $$\frac{\text{d}^2y(x)}{\text{d}x^2}=0$$ where $$\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$$ with $$y(0)=y_0$$ and $$a,b,c,d>0$$ If it helps I can ...
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0answers
30 views

Definition of omega limit set

We say that $p$ is an omega limit point of $x$ if there exists a sequence $\{t_n\}, t_n \rightarrow \infty$ such that the flow $\pi(t_n,x) \rightarrow p$. The set of all such points is called the ...
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0answers
28 views

step by step solution of $y''+(x-1)y=0$ by Frobenius method

I've tried solving this equation, $y''+(x-1)y=0$, but honestly I'm not sure if I've done it right. Using Frobenius Method, I got $r_1=1$ and $r_2=0$ as the indicial roots which I think under case 3 as ...
2
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2answers
56 views

Which method is correct?

To solve this equation $X''+k^2X=0$ we look to the solution in the form $X(x)=e^{rx}$ which has roots $r=\pm i k $ My question is? If we need to apply the BCs $X(\pm l)=X(0)$ which one of the ...
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1answer
67 views

How can I prove $x^3\, \frac{d^3 y}{dx^3} = \Delta(\Delta-1)(\Delta-2)y$?

This equation is used to solve Cauchy Euler Equation As it can be seen author has provided explanation of the fact how ...
2
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0answers
24 views

In the Glycolysys Sel'kov model, what are the meaning of “a” and “b” values?

In the Sel'kov model of glycolysis wich I put on next $u'=-u+av+u^2v\\ v'=b-av-u^2v$ wich have a limit cycle and have all sense because it is a glycolytic cicle. What are the ...
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1answer
21 views

Why does Abel's identity imply either $W = 0$ or $W \neq 0$ everywhere?

Let $y_1$ and $y_2$ be solutions to the linear differential equation $A(x)y'' + B(x)y' + C(x)y = 0 $ and let $W = W(y_1, y_2)$ be the Wronskian of the solutions. Why does Abel's identity ...
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0answers
17 views

Computing $\bigtriangledown r^m$ knowing the vector $\boldsymbol r$

I am asked to compute $\bigtriangledown \cdot \boldsymbol r$ and $\bigtriangledown r^m$ for $m$ constant, where $\boldsymbol r =x \boldsymbol i+ y\boldsymbol j +z\boldsymbol k$ and $r= |\boldsymbol ...
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1answer
34 views

Differential equations - approach for certain types of questions

Problem: $$(\frac{dy}{dx})^2 - x\frac{dy}{dx} + y = 0$$ I attempted to solve the equation by assuming $\frac{dy}{dx}$ to be $t$. I then used the formula for general solutions of a quadratic ...
1
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1answer
22 views

particular solution of non-homogeneous differential equation

If we assume $A e^{-t}$ (method of undetermined coefficient)to be a particular solution of $y'' - 3y' - 4y = 2e^{-t}$, we can't find the $A$. The book says to try with $Ate^{-t}$. I am trying to ...
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1answer
59 views

Differential Equations first order [closed]

Anyone who can help me on this equation, $y' = (\frac{y}{x + y^3})$ I've already tried to make a substitution which is: $h(x,y) = x+y^{3}$ and did the derivatives but still no solution, so if anyone ...
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0answers
39 views

Stability of origin of dynamical system

Usually you can note some nice structure in the problem which enables construction of a nice Lyapunov function. But this one is just a monster. Maybe there is a trick I've missed? Investigate the ...
4
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0answers
62 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
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2answers
37 views

Find the general solution and draw the phase portrait

Find the general solution and draw the phase portrait for the following linear system: $x_{1}^{\prime}=x_1$ $x_{2}^{\prime}=x_2$ My Procedure: The method of separation of variables can be used to ...
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1answer
25 views

Multiscale analysis with non-integer exponents

I am dealing with the following non-linear differential equation: $$\frac{d^2 x}{d t^2}=2\varepsilon\frac{d x}{d t}-\left(\frac{d x}{d t}\right)^3-x$$ I found that $x=0$ is the only one fixed point ...
2
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1answer
78 views

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
1
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1answer
20 views

Scale invariant ODE. Is this general method correct?

Recently, a question I asked had the differential equation $y''=xyy'$. A trick to solving this quickly is to notice that scaling $y$ by $a$ and $x$ by $b$ shows that $a=1/b^2$ is the condition that ...
4
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1answer
45 views

Slightly different results to an ODE system - hand calculation vs Mathematica

This has been driving me mad for the last few days. I have a a pair of ODEs: $$\frac{d^2 M_N}{d x^2}=\lambda_{N}^2 M_N$$ $$\frac{d^2 M_{N-1}}{d x^2}=\lambda_{N-1}^2 M_{N-1}-\frac{f}{d_{N-1}}M_N$$ ...
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1answer
42 views

How can I find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$
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1answer
22 views

Building the foundations of DE

I have seen tons of times that $f(x)=f'(x)$ implies $f(x)=Ce^x$. But the "proof" involves a division by $f(x)$. My question: Suppose that $f:\Bbb R\to \Bbb R$ is a continuous, differentiable ...
4
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1answer
45 views

A question on ordinary differential inequality

Could we find a solution $f=f(x)$ to the following initial problem for the OD inequality? $$3xf'+f-\sqrt{6f}\leq 0,\quad f(0)=0,\quad f(8/3)=6.$$ . Added: The above question is in fact a special ...
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0answers
16 views

A mixing problem with concentrations

These mixing problems trip me up sometimes and I was just wondering if my setup was correct. It asks: A tank with a capacity of 500 gallons originally contains 200 gallons of water with 100 lb. of ...
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1answer
15 views

Existence of solution to second order linear PDE

Suppose $f$ is a given smooth function on $\mathbb{R}^2$. I want to show that for $a,b,c \in \mathbb{R}$ such that $b^2 - ac > 0$ there exists a smooth function $u$ such that $$ a\frac{\partial^2 ...
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1answer
27 views

Lipschitz condition (coordinate choice) [closed]

Why the Lipschitz condition is defined with respect to one variable in plane? I have only seen the cases where this condition is w.r.t. $y$ coordinate. Can it be w.r.t. $x$ coordinate? ${}$
3
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1answer
53 views

Solving linear differential equations

Find the general solution for the following equation: $$\frac{dy}{dt}+2ty=\sin(t)e^{-t^2}$$ Find a solution for which $y(0)=0$ First I found the integrating factor which is $e^{t^2}$ ...
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0answers
11 views

book for parabolic partial differential equation

I need the book "Linear and quasilinear equations of parabolic type" By Olʹga Aleksandrovna Ladyzhenskai͡a, Vsevolod Alekseevich Solonnikov, Nina N. Ural'tseva.The price range of the hard copy is ...
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0answers
13 views

Differential equation initial guesses

I am using Matlab to solve for differential equation boundary value problem. (bvp4c) However, I am at a completely loss when it comes to choose a initial guess for y and y'. I realize that this ...
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0answers
23 views

how to solve differential equations using Gear's BDF(Backward Difference Formula) method

Hi i am trying to solve coupled stiff differential equations in c++. I used Euler and RK methods but it is giving only few values , after that it is giving Nan values. I tried with C++ libraries also ...
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0answers
20 views

Existence of solution to linear second order PDE.

Suppose $f$ is a given smooth function on $\mathbb{R}^2$. I want to show that for $a,b,c \in \mathbb{R}$ such that $b^2 - ac > 0$ there exists a smooth function $u$ such that $$ a\frac{\partial^2 ...
0
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1answer
19 views

Determine the order of consistency of $y_{n+1}=y_n+(h/2)(y_n'+y_{n+1}')+(h^2/12)(y_n''-y_{n+1}'')$ (I want to improve my answer)

I can solve this problem but I was wondering if there is a quicker way to do it since time will be tight during the exam... I would really appreciate your tips and advice on how to calculate this in a ...
4
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2answers
45 views

How does the PDE $\,\dfrac{d^2u}{dx^2} = 0\,$ become $\,u=x\,f(y)+g(y)\,$ when integrated?

Given that $u(x,y)$ can someone please explain to me how the result as asked in the question is achieved? Steps would be really appreciated, thanks.
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0answers
63 views

Metric-space complete?

My question is if a specific metric-space is complete, respectively under which conditions it is complete. I am rather a newby, but hope that the question is understandable. The metric-space is ...
2
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0answers
30 views

please help me to find the soluton of the following 2nd order ODE [closed]

Equation is $y′′(t)−(A/t)y′(t)−By(t)=0$ please find help me to find out the analytic solution I applied all general method but not able to solve it.
2
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2answers
53 views

Application of Poincaré-Bendixson theorem

Consider the system $$x' = 3xy^2-x^2y \\ y' = 5x^2y - xy^2$$ Show that the system has no periodic solutions. This is a tricky example. Linearization leads nowhere and I'm having a hard time ...
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0answers
43 views

Dirac delta - $\delta(t-S)$ - impulse function at multiple occasions S

Apologies in advance, my mathematics is likely to be very ad hoc, but I hope it makes sense... I have a software package that models data using multivariate stochastic differential equations, however ...
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1answer
40 views

Expotential Growth/Decay - Problem Deriving Atmospheric Pressure Formula

I have a problem deriving the following formula: $$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$ Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ ...
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1answer
52 views

Initial value problem, not sure where to begin!

Show that the function $y(t)=t^2$ satisfies the initial value problem $\frac{dy}{dt}=2\sqrt{y}, t\geq{0}; y(0)=0$ Show that this initial value problem does not have a unique solution, by ...
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1answer
19 views

Solving homogeneous linear DE of n degree using Wronski determinant

Here is my task: Explain use of Wronski determinant on solving homogeneous linear DE of n degree: $y^{(n)}=a_{n-1}(x)y^{(n-1)}+...+a_1(x)y'+a_0(x)y$ Any idea?
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1answer
34 views

Find the characteristic equation in terms of $p$ rather than $\lambda$ in second order differential equation?

Question Consider the following second-order differential equation with constant coefficients, $$y''\left(x\right)-10\,y'\left(x\right)+41\,y\left(x\right)=0$$ By seeking solutions of the form ...
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1answer
17 views

Ordinary Differential Equations by Morris Tenenbaum and Harry Pollary

On definition 2.68, the book states that a set in the plane is called a region if it meets two conditions (p. 14): "Each point of the set is the center of a circle whose entire interior consists of ...
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0answers
58 views

Is it possible to bruteforce a differential equation

Is there any method to solve differential equations which involves just a number of basic functions combined into various permutations (with various factors) which are then fed into the differential ...
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0answers
36 views

Find the integrating factor and solve for the equation.

$\left(2xy^2-y\right)dx + \left(2x-x^2y\right)dy= 0$ $2\,dx + \left(2x-3y-3\right) dy = 0;\quad y\left(2\right)= 0$ $\left(2y\sin\left(x\right)+3y^4\sin\left(x\right)\cos\left(x\right)\right)dx - ...