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

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2
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1answer
126 views

A particular version of Gronwall's inequality

We have this theorem (Gronwall's inequality): Let $f$, $g$ and $h$ be continuous nonnegative functions defined for $t\ge t_0$. If$$f(t)\le h(t)+\int_{t_0}^{t}g(s)f(s)\,ds\>,$$then$$f(t)\le ...
1
vote
0answers
58 views

Solve using Riccati equation?

In this problem, they've asked to solve it using the solution $y=-\frac{1}{3x}$ but the problem is that the solution just makes it more complicated. The equation which needs to be solved is, ...
1
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0answers
88 views

How to show that the first derivative is bounded in a function

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation} How to show in above function the ...
1
vote
1answer
105 views

Integration Factor in terms of $x$ and $y$ used to reduce an ODE to an exact form

If your ODE is non exact you need an integrating factor to make it exact I'am asking for a general expression for finding this integrating factor Let the Integration factor be $T(x,y)$ ...
0
votes
1answer
38 views

Let A and B are matrices such that all eigenvalues ​​have negative real part. Flows of the systems $X ' = AX$ and $X ' = BX$ are conjugated?

As the eigenvalues ​​are negative, we have that the equilibrium points are the way to sink thus suggests that they are conjugated.
1
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2answers
342 views

Diffeomorphism in $\mathbb{R}^n$

I am stuck with this question for quite some time now. Please help. Let $f$ and $g$ be $2$ linearly independent vector fields on $\mathbb{R}^n$. State with reasons if one can always get a ...
1
vote
2answers
151 views

Derivative of mixed matrix terms with inverse matrix

I've been trying to solve two matrix derivative terms including an inverse matrix but I am unable to find a clue : 1) Derivative of $KG^{-1}J$ with respect to $G$. 2) Derivative of ...
2
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2answers
69 views

Remembering Special Function Equations?

I have an awful memory when it comes to factoids, I need to remember the Legendre, Hermite, Laguerre, Chebyshev, Hypergeometric & Jacobi equations, all of which are of the form $p(x)y'' + q(x)y' + ...
1
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2answers
56 views

Solving a PDE - do you have an idea?

Do you have an idea how to solve $$ v_{\xi\eta}=\frac{1}{2} v_{\xi}\cdot\xi? $$ First I thought of using $$ v_{\xi\eta}=v_{\eta\xi}, $$ substituting $z:=v_{\xi}$ and then getting $$ ...
0
votes
2answers
32 views

Openness of the set of hyperbolic linear systems

I need to prove that the set $S=\{A\in M_n(\mathbb{R}); x'=Ax\hspace{0.2cm} is\hspace{0.2cm} hyperbolic\}$ is open in $M_n(\mathbb{R})$ the set of real matrices of order $n\times n$. I have already ...
1
vote
1answer
229 views

Fourth order differential equation

I have this physics mathematical problem : (see link in comment) $$EI \frac{∂^4u}{∂x^4}= f \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ The boundary conditions are: ...
2
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1answer
357 views

Undamped forced motion and resonance, find $ω$ given an external force

The position of a certain spring-mass system satisfies the initial value problem $$6x''+kx=0,x(0) = 5,x'(0)=v$$ The period and amplitude of the resulting motion are observed to be $3π$ and $6,$ ...
0
votes
1answer
60 views

Equating coefficients $A-2B\sin x=2-\sin x$

I'm trying to find out how to find $A$ and $B$ for the equation $A-2B\sin x=2-\sin x$ I know I'm supposed to get $A=2$ and $B=\frac{1}{2}$, and I've looked on Google for help but didn't understand ...
1
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1answer
90 views

Solve second order ODE with undetermined coefficients method

Consider the differential equation $$y''+5y'=-sin(x)-1$$ Find the general solution. Here's my work: I found the solution to the homogeneous equation to be: $y_h(x)=C_1e^{-5x}+C_2$ And for the ...
7
votes
3answers
209 views

Differential equations solveable independently of coordinate system?

Looking from a physics viewpoint ODEs tend to look very differently when setting up the problem in different coordinate systems. For instance the Laplacian in spherical coordinates involves way more ...
1
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2answers
50 views

None exact first order ODE

i have to solve the following $1^{st}$ order differential equation $(xy+1)dx+(2y-x)dy=0$ i am in the elementary differential class,and have not learned multivariate functions, the equation below is ...
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vote
2answers
617 views

Find a second order ODE given the solution y.

Find a second order differential equation so that $$y=C_1e^{-3x}cos(4x)+C_2e^{-3x}sin(4x)+4e^{3x}$$ solves the differential equation for any choice of $C_1$and $C_2$ The answer should be in the ...
1
vote
1answer
107 views

Understand Picard-Lindelöf Proof

I am trying to understand the Picard-Lindelöf from my book which uses the fixed point theorem. The task is trying to find $x \in C(a,b)$ in open interval $(a, b)$ containing $t_0$ such that it ...
2
votes
1answer
183 views

Forced nonlinear oscillator - analytical methods

This is an example from Kovacic & Brennan (2011). Consider the following equation of motion for a forced, non-linear oscillator (Duffing's equation): $$ \ddot x + 2 \zeta \dot x + \alpha x + ...
0
votes
1answer
70 views

solution of a system

Let $A$ an $n \times n$ matrix and let $B(x)$ a continuous $n \times n$ matrix, and for $x \geq 0$ and a vector $Y$ with $n$ components, we consider the system $$Y' = (A+B(x))Y\tag{2}$$ If all ...
0
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1answer
249 views

Find a particular solution for second order ODEs using undetermined coefficients method

Match the appropriate form of the particular solution labelled A through J with the differential equations below. Enter K if all of the particular solutions are incorrect. $$y''-5y'-24y = 3xe^{2x}, ...
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vote
1answer
104 views

Differential equation application question

The air in a room with volume $200m^3$ contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of $2m^3/min$ and the mixed air flows out at ...
0
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1answer
118 views

ODE system and show infinite number of positively invariant ellipsoids

The system of ODEs is: $$ \dot{x} = -2x+yz \\ \dot{y} = x-xz \\ \dot{z} = xy $$ I found two lines of equilibria etc. but I now need to find the parameters for this "energy" or Lyapunov function, so ...
2
votes
1answer
279 views

$\frac{dy}{d \theta} = {e^y\sin^2(\theta)\over {y\sec(\theta)}}$

Please help me solve the above differential equation. I'm confused as to the steps required to obtain the answer
3
votes
1answer
70 views

$\frac{dx}{dt} = |x|^{1/2}$

Im looking to find 4 solutions to the ODE : $\frac{dx}{dt} = |x|^{1/2} , x(0)=0$. Clearly, $x=0$ is one solution. Using seperation of variables for $x>0$ yields $x= t^2/4$ as another solution, ...
7
votes
2answers
6k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
2
votes
2answers
122 views

Changing coefficients in second order ODE

I am considering the following equation: $$z''(t)+Az'(t)+B=0$$ I need to reduce it to the form $$u''(t)+u'(t)+1=0$$ by "a linear change of variables z,t" what do you think it means? I have tried ...
2
votes
2answers
73 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
3
votes
3answers
345 views

Differential equation: autonomous system

This isn't homework. I have no idea what theorems I should be looking at to solve this. Guidance, partial and total solutions are all welcomed. Let $f$ be a locally lipschitz function in an open ...
0
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1answer
54 views

Significance of the DEGREE of Differential equation

Can anybody gives the idea why degree of a differential equation is important?. Every differential equation book writes the definition of "degree of a differential equation". But, why do we care about ...
3
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2answers
324 views

how do I solve this seperable equation with so many terms?

Solve given differential equation by separation of variables $$\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}$$ I started by multiplying each side by the denominator to get $$(xy-2x+4y-8) dy = ...
3
votes
2answers
68 views

are all dynamical systems described by differential equations?

we defined in lecture a dynamical System as a one-parameter family of maps $\phi^t:M\rightarrow M$ such that $\phi^{t+s}=\phi^t\circ\phi^s$ and $\phi^0=Id$, where $M$ is some (smooth) manifold and ...
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0answers
123 views

Monodromy Groups of Differential Equations

I have heard that monodromy groups and analytic continuation can be used to construct new solutions to a differential equation from a particular solution. What references (textbook, or papers) could I ...
1
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1answer
102 views

Maximal unique solution to an IVP.

In class we learned the existence and uniqueness theorems for differential equations. The weaker Picard-Lindelof states that for any IVP, $$ \begin{cases} x'(t) = f(t, x(t))\\ x(t_0) = x_0 \end{cases} ...
0
votes
1answer
58 views

How do I solve $y'''-5y''+11y'-15y=0$?

How do I solve the following linear ordinary differential equation with constant coefficients? $$y'''-5y''+11y'-15y=0.$$ Please help. Thank you.
4
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1answer
64 views

Gaussian density function satisfies $y'=-xy$. Coincidence?

Is part of the rationale for the Gaussian distribution that the density function satisfies the differential equation $y' = -xy$? Or is this more or less incidental?
0
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1answer
100 views

Normalizing (Non-Dimensionalising) the Young-Laplace equation

I have a simple fluid statics problem of a liquid drop, resting on a stationary flat solid surface with a static gas of constant pressure above. The density in both the gas and liquid are constant. To ...
3
votes
1answer
72 views

Proving that $\frac{d}{dt}x=\sqrt{|x|}+a$ has unique solution

Can you help me with this question, please : Given $a>0$, I want to show that $\frac{d}{dt}x = \sqrt{|x|} + a $, with $x(0)=x_{0}$, has an unique solution. The existence is granted by Peano's ...
0
votes
1answer
43 views

linear ode -order n

with the formula of solution of an order $n$ linear differential equation, calculate the exact solution of the differential equation: $$y'''-3y'+2y=9 e^x, x>0$$ step 1: found the exact solution of ...
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4answers
86 views

What's the integral of $\frac{-4x}{1+2x}$?

Calculate the following: $$\int \frac{-4x}{1+2x}\ dx$$ I got $-1-2x+\ln(1+2x)$ as a result. But why does the answer say it is just $-2x+\ln(1+2x)$? Where did the $-1$ go? Thank you
2
votes
1answer
68 views

solve this equation $z(z+y)dx+z(z+x)dy=0$

I need to solve this following equation $$z(z+y)dx+z(z+x)dy=0$$ I get this from above equation $$\frac{dx}{z(z+x)}+\frac{dy}{z(z+y)}=0$$ After there, I dont know what I need to do.
0
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1answer
73 views

Find solution(primitive) of the equation

I want to find its solution of the following equation $$ydx+xdy+2zdz=0$$ answer: Keeping $z$ constant; I obtain that $$ydx+xdy=0$$ or $$\frac{dx}{x}+\frac{dy}{y}=0$$ Then I get $$U(x,y,z)=xy$$ ...
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1answer
87 views

ODE Theory: Are centers and linear centers the same for reversible systems?

everyone! I'm trying to prove that a linear center of a planar system IS a center when the system is reversible (invariant under the change of variables $t\mapsto -t$ and $y \mapsto -y$). I fooled ...
0
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2answers
64 views

Ordinary Differential Equation how to solve this

Hi i need to learn in fast way how to solve ode i still have problem with this. I need to find $y = y(x)$ having $y′ = 0$, $y′′ = 0$, $y′′′ = 0$, $y'''' = 0$ I am guessing that $y= e^{rx}$ Then we ...
2
votes
1answer
41 views

Help me Verifying that the equation is integrable and finding its solution

How can I verify that the equation is integrable and that find its solution; $$2y(a-x)dx+[z-y^2+(a-x)^2]dy-ydz=0$$ Honestly, I tried too much, but I got too strange results,thus I couldnt show my ...
0
votes
1answer
32 views

Solution to simple inhomogeneous differential equation

What's the solution to this differential equation? $\frac{\mathrm{d}u(t)}{\mathrm{d}t}+u(t)=\sum_i\delta(t-t_i)$ Intuitively I would say it's $u(t)=\sum_i\mathrm{e}^{t_i-t}\sigma(t-t_i)$ where ...
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1answer
59 views

How to obtain the last ratio $\frac{d(x+z)}{(x+z)}$

I am studying example-2.3 In the first line, it says that "the numerators and denominators in the first and last ratio" And the following is obtained $$\frac{d(x+z)}{x+z}=\frac{dy}{y}$$ But I ...
2
votes
1answer
107 views

How to solve this ODE equation $\frac{dy}{dx}=\frac{y^6-2x^2}{2x^2y+2y^3-y}$?

solve this ODE equation $$\dfrac{dy}{dx}=\dfrac{y^6-2x^2}{2x^2y+2y^3-y}$$ My try: $$\dfrac{y~dy}{dx}=\dfrac{y^6-2x^2}{2x^2+2y^2-1}$$ let $$u=y^2$$ then ...
1
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1answer
65 views

Existence of a global solution to $y = f(y,x)$ when $f$ is continuous

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be continuous. By Peano theorem there exists a local solution to the Cauchy problem $$ \begin{cases} y' = f(y,x),\\ y(0) = y_0. \end{cases} $$ If I ...
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0answers
71 views

1D PDE decoupling

I need to solve the 1D nonlinear poisson equation and I thought of trying the fixed point decoupling technique. The equation is this: $\frac{\partial^2 \phi(y)}{\partial ...