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

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0
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1answer
57 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 ...
3
votes
0answers
37 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, ...
0
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0answers
55 views

how did he turn from that from to that from ? is that possible or wrong mathematically [closed]

$$U=e^{x+\ln x}= (e^x)(e^{\ln x})= (\ln x)e^x$$ How did he turn it from $(e^x )(e^{\ln x})$ to $(\ln x ) e^x$
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2answers
30 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 ...
1
vote
1answer
22 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
votes
0answers
33 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
vote
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
votes
1answer
43 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: $$\frac{d^2 M_N}{d x^2}=\lambda_{N}^2 M_N$$ $$\frac{d^2 M_{N-1}}{d x^2}=\lambda_{N-1}^2 M_{N-1}-\frac{f}{d_{N-1}}M_N$$ ...
0
votes
1answer
41 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
votes
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 ...
1
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0answers
14 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 ...
0
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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 ...
0
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1answer
24 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
votes
1answer
52 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 $e^{t^2}$ ...
0
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0answers
9 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 ...
0
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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 ...
0
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0answers
19 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
votes
2answers
43 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.
0
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0answers
53 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
29 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
votes
2answers
52 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 ...
1
vote
0answers
42 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 ...
0
votes
1answer
39 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 $\,A = A_0\left(e^{\,kt}\right)\,$ ...
0
votes
1answer
51 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 ...
0
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1answer
17 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?
0
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1answer
33 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 ...
0
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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 ...
2
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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 - ...
-1
votes
1answer
22 views

Find the fixed points of the following dynamical system

Find the fixed points of the following dynamical system\begin{align}\frac{dx}{dt}&= (a_1 -b_1x - c_1y)x \\ \frac{dy}{dt} &= (-a_2 +c_2x)y\end{align} Note that ALL the parameters are ...
3
votes
1answer
41 views

Perpendicular Gradients

Suppose $f:\mathbb{R}^2\to \mathbb{R}$ is smooth. Further suppose $\nabla f$ vanishes no where. When is it possible to find a smooth non-singular $g:\mathbb{R}^2\to \mathbb{R}$ satisfying $\nabla ...
0
votes
1answer
45 views

General integral of $y' = 2t\sqrt{1 - y^2}$

I have doubts about the general integral of $y' = 2t\sqrt{1 - y^2}$. This is my attempt of solution: The equation $y' = 2t\sqrt{1 - y^2}$ is of the form $y' = a(t)b(y)$ so I try to solve it by ...
1
vote
2answers
27 views

Simple coupled ODE

Find two linearly independent solutions to the pair of coupled ODEs $\frac{dx}{dt} = 2x + 3y$, $\frac{dy}{ dt} =-3x+2y$. I figured the eigenvalues/eigenvectors of the corresponding matrix to be ...
0
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0answers
20 views

Solving $\frac{d^2 z}{dr^2}=\left(\frac{1}{c(r,z)}\ \frac{\partial c(r,z)}{\partial z}\right)\cdot\left[1+\left(\frac{dz}{dr}\right)^2\right]$

Solve the following differential equation $$\frac{d^2 z}{dr^2}=\left(\frac{1}{c(r,z)}\ \frac{\partial c(r,z)}{\partial z}\right)\cdot\left[1+\left(\frac{dz}{dr}\right)^2\right]$$ where $c$ is a ...
0
votes
1answer
43 views

Derivative of integral $\int_0^{\infty} e^{-x \cosh t} dt$

I am given the integral $y = \int_0^{\infty}e^{-x\cosh t} dt$ and wish to show that this integral solves the modified Bessell equation: $x^2y'' + xy' -x^2y = 0.$ To do this I need to calculate the ...
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0answers
37 views

Homogeneous linear DE with constant coefficients

Here are some question from book, I have to answer with "yes" or "no": a) $\phi(x)=x^{2}+3$ is solution of some homogeneous linear DE with constant coefficiens, b) $\phi(x)=\frac{1}{2+x^{2}}$ is ...
0
votes
2answers
32 views

Where is the error in my reasoning about this first-order linear differential equation?

Considering this first-order linear differential equation: $\frac{dy}{dx} + 2y = 0$ Although I now know the correct general solution to be $y = c_1e^{-2x}$, I cannot figure out where I am going ...
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0answers
24 views

Sturm-Liouville Problem: Possible range for the eigenvalues.

Let us consider the Sturm-Liouville Problem $$y''=\lambda \cdot y$$ with $y(0)=y(\pi)=0$. As only $\lambda \gt 0$ renders non-zero solutions, one obtains the condition $\sin(\mu \pi)=0$ for a ...
0
votes
1answer
48 views

Continuously Differentiable in $\mathbb{R^2}$

I understand the concept of continously differentiable (first derivative is continuous) in $\mathbb{R}$, however what does it mean for the RHS of: $\dfrac{d}{dt} ...
1
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3answers
51 views

ODE, a first order none-linear equation with ratios and exponents

I have this equation that I don't know how to approach: $$xy'-y=xe^{\frac{y}{x}}$$ I don't even know if the solution is a function of variable $x$ or $y$. I don't know how to start. Any ideas?
1
vote
0answers
19 views

Find the critical curves for the following functional with subsidiary conditions

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1}\left(y'^2+z'^2-xyz'-yz\right)dx$$ with subsidiary conditions : $$\int_{0}^{1}\left(y'^2-xy'-z'^2\right)dx=2$$ ...
0
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1answer
26 views

Parameterizing a cycloid, and finding the arclength

I want to: 1) Parameterize a curve of a cycloid passing through the origin, of a disk of radius $1$, 2) Calculate the arc-length of one cycloid corresponding to one full rotation of the disk. ...
2
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2answers
39 views

Elementary Differential Equations

I'm currently studying Elementary differential equations, and I came across a confusion that I had that I think arises from notation, but I would like to clarify with someone. The example problem said ...
0
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1answer
43 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a ...
6
votes
0answers
58 views

Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
-2
votes
2answers
25 views

Find homogenous linear second order DE if solution is known [closed]

My task is to find homogenous linear second order DE if its solution is $y(x)=C_1x^{2}+C_2x^{4}$. Any idea?