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Are these equations "properly" defined differential equations?

1) Are all these differential equations with terms $(T-t)$ "well-defined" as differential equations? Their well-posenedness will depend on the multiplicity of the singularities. You will ...
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Differential equation (encyme reaction)

The sets of conserved species can be found by looking at the left null-space of the stoichiometric matrix $S$ given in this case by $$S=\begin{bmatrix} - 1 & 1 & 0 & -1 & 1 & 0 &...
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Euler integration solution from system of ODE's - already estimated values

Ok, I think I see what you want. We have, $$\frac{d I}{d t}= \beta V \left[ 1- \frac{I}{S} \right] - k_a I \frac{A_3^2}{A_3^2+g}$$ In functional form, $$ \frac{d I}{d t}(t) = F(V,I,S,A_3)$$ What you ...
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1 vote

Simple eigenvalues in Sturm-Liouville problem

I'm a student myself and I only just started a course on this topic so my method may not be the most rigorous or fully thought-out but if we assume two solutions, U and V that satisify the sturm-...
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2 votes

Quadratic diferential equation

This is separable. Solve \begin{gather*} \boxed{u^{\prime}-C \left(v^{2}-2 u v +u^{2}\right)+k=0} \end{gather*} In canonical form the ODE is \begin{align*} u' &= F(t,u)\\ &...
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5 votes

How many functions are possible for the given differential equation?

My answer is treating $y'=2\sqrt{|y|}$ instead of $y'=2\sqrt{y}$. In my experience the problem I consider is the more common way this exercise is stated. If you actually meant $y'=2\sqrt{y}$ and are ...
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An inequality for a maximal solution of an IVP

You have $$ |y'(x)|\le|x|. $$ The upper bound follows directly, the lower bound from the sign of $f$.
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1 vote
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Solution of 2nd order ODE

Let $A$ be a primitive of $a$, then $$ \frac{d^2}{dx^2}A(u(x))=\frac{d}{dx}a(u(x))u'(x)=a(u)u''(x)+a'(u)u'(x)^2. $$ This fits exactly the first two terms of your equation, so $$ A(u(x))=-\frac{b}{2}x^...
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How to prove $A\cos(\omega t-\phi)$ = $a\cos(\omega t)$ + $b\sin(\omega t)$ using $e^{i\theta}$?

To calculate $A$ and $\phi$ from $a$ and $b$: $$ A = \sqrt{a^2+b^2} $$ $$ \phi = \arccos(\frac{a}{A})=\arcsin(\frac{b}{A}) $$
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1 vote

Input for stopping in Dubin's path dynamics?

I don't think your question is dumb, but the design of the ODE system is quite dumb. The speed for moving along $x,y$ plane is a constant, $v_s$, while $u$ only controls rotation. In this design, ...
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1 vote
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Solution (recurrence relation) of non-linear DE using the method of power series

The differential equation to be solved is $$ y' -e^y -x^2 = 0 ,\qquad y(0)=c $$ and the suggested approach is with power series. It may not be immediately obvious, but after working with the power ...
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a question on visualizing a state of DAE and a question from a continous time nonlinear dynamics

The first kind is the classic ordinary differential equation (ODE), the 2nd is a generalized version of it. The matrix $E$ is sometimes called the mass matrix (in analogy of Newtonian Mechanism). In ...
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2 votes
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fourth-order finite difference for $(a(x)u'(x))'$

Let's use $$ f'(x) = \frac{f_{i-3/2} - 27 f_{i-1/2} + 27 f_{i+1/2} - f_{i+3/2}}{24 \Delta x} + O(\Delta x^4) $$ as an approximation for the outer derivative. To obtain a fourth-order approximation for ...
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1 vote

What is the solution to $y'=p(x)+y $ where $p(x) =\sum_{k=0}^d p_kx^k $?

It suffices to solve $y'=y+x^n$ for any $n,$ multiply by constants, and add up results. The homogeneous equation $y'=y$ admits solutions of the form $ce^{x}.$ Next we should look for a particular ...
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2 votes
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Power series solution of differential equation $y'=x^2 +y$

I think the series in your answer is correct, but what is the question? If you want an abbreviation here is one $$ 1+x+\frac12(x^2+x^3)+\sum_{k=4}^\infty\frac{3}{k!} x^k \\=3\sum_{k=0}^\infty\frac{1}{...
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1 vote

Solution (recurrence relation) of non-linear DE using the method of power series

The power series for $e^y$ can be computed via the differential equation of the anti-logarithm. Wtih $v=e^{y}=\sum b_nx^n$ one has $$ v'=vy'\implies \sum nb_nx^{n-1}=\sum b_m\,(n-m)a_{n-m}x^{n-1} \\~\\...
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6 votes

An interesting recurrent equality, possibly easier to solve in its differential form?

Your question comes in two variants. One is discrete. The other continuous. The discrete variant asks about the equation $$ f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m). \tag{1} $$ A nicer version of this ...
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1 vote

An interesting recurrent equality, possibly easier to solve in its differential form?

Being a combination of product and sum, it looks difficult that there might be a closed form. The best I can suggest is to make the substitution $$ f(n) = 2^{g(n)} $$ I am using $2$ as a base because ...
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construct different ode systems but with the same lyapunov function

Let us consider a family of Hurwitz stable matrices $\{A_1,A_2,\ldots\}$ which may be finite or even infinite. A result by Liberzon, Hespanha, and Morse proved that it there is basis change $Q$ such ...
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2 votes

Determining a function which fits the given differential equation

Let $f(x)=\frac{1}{2}x$. Then $$f'(x)=\frac{1}{2}$$ and $$\frac{\int_0^x f(t)\,\mathrm{d}t}{xf(x)}=\frac{\frac{1}{4}x^2}{\frac{1}{2}x^2}=\frac{1}{2}.$$ Can't tell you how I came up with it though, I ...
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1 vote
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How can we derive $u_0 (t) = a_0 \cos (\omega_0 t + a_0)$ from this solution?

You misread the paper slightly, there are two constants in the general solution, $$ u_0(t) = a_0\cos(\omega_0 t + \underset{\uparrow}{\alpha_0})$$ This is equivalent to the form you got by the angle ...
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"Spanning" of solutions of ordinary differential equations

It is not true in general that the spanning of the switched system can be obtained from the spanning of the individual systems. Actually even if all individual systems are asymptotically stable the ...
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Proof of Wronskian relation using induction

I'm not sure if I have to use induction method. If you are allowed to use Jacobi's formula things will be even more straight forward. $$ \frac{d\det X(t)}{dt}=\det X(t) tr(X(t)^{-1}\frac{dX(t)}{dt})\\ ...
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Bootstrap proof of Gronwall's inequality (exercise 1.26 in Tao)

I think your proof is correct, but it can be made simpler by first observing that $$\int_{t_0}^t B(s)e^{\int_{t_0}^sB(s')\,ds'}\,ds=e^{\int_{t_0}^tB(s)\,ds}-1$$ The inequality $F(t)\leq G(t)$ is then ...
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1 vote
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How to show that the solution of a delayed differential equation is unique? Can you use Picard Lindelof theorem?

For this example, you do not need much. You can apply what is called forward propagation (or the method of steps). First consider $t\in[0,1]$. In that case, we have that $$y'(t)=\alpha y_0(t-1),$$ and ...
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Numerical Differentiation Table

If $s(t)$ is travelled distance at straight line motion, then speed is $s'(t)$ and acceleration is $s''(t)$. For numerical differentiation there are various formulae. Most of them can be obtained ...
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1 vote
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linearization of non-linear ODE

You might want to read up on the Schur-complement. Or linearize the DAE system directly \begin{align} \dot u &=D_xf(x^*,y^*)u+D_yf(x^*,y^*)v,\\ 0&=D_xg(x^*,y^*)u+D_yg(x^*,y^*)v, \end{align} $y^...
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1 vote
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What is a sublinear function? Is $y'(x) = x^5(e^{4-y^2}-1)$ sublinear?

Yes. It is even bounded in $y$. So you get $$ |y'|\le |x|^5(e^4-1) $$ which implies $$ |y(x)-y(0)|\le \frac{|x|^6}6(e^4-1). $$ This is the kind of bound one can use to show that the maximal domain is $...
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1 vote
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Examples of dynamical systems that have structural stability

Anosov diffeomorphisms (= diffeomorphisms whose hyperbolic set is the whole state space) are the classical examples of structurally stable diffeomorphisms (the reason why such diffeomorphisms are ...
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1 vote
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Examples of hyperbolic sets of dynamical systems

Here are some standard examples: Hyperbolic fixed points of a diffeomorphism: Let $f:M\to M$ be a $C^1$ diffeomorphism of a $C^\infty$ manifold and $x\in M$ be such that $f(x)=x$. Then the derivative ...
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2 votes

Using matrix to analyze an ODE system $\begin{cases}\dot a_k(t)=2(b_k^2-b_{k-1}^2)\\\dot b_k(t)=b_k(a_{k+1}-a_{k})\end{cases}$ with $b_0(t)=b_n(t)=0$.

These are the equations of motion of the Toda lattice . It is an integrable classical mechanical system, with a Lax matrix given by $L(a,b)$ as you define. The eigenvalues of a Lax matrix can be shown ...
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solve the pde without any initial condition $xu_x-xyu_y=u$

If you eliminate $t$ for $x$, you get $u=c_1x$ and $y=c_2e^{-x}$. The idea now is that the family of characteristics in a solution surface only has one parameter, so the constants have a dependency. ...
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2 votes

Proper ways/strategies of observing and transforming ODE / PDE

That first one is not that bad; in particular you have probably seen $x''+x=0$ before, which has the same basic feature. Basically if there's a periodic solution then there must be a conserved ...
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4 votes

To find degree of following differential equation $(y^")^{1/2}=(y^")^5$.

The order of a differential equation is the order of the highest derivative that appears in the equation. Since $$(y\color{red}{''})^{1/2}=(y\color{red}{''})^{5},\quad y=y(x)$$ so you have a second ...
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1 vote

Question with ODE's without constant coefficients

You can combine terms as $$ x^2x''-(x+2)(xy'-y)=2x^3. $$ From this it can be seen that $y=x$ is a basis solution of the homogeneous equation. The equation should thus simplify when trying to find the ...
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0 votes
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Deduce stability from strict convexity of gradient systems

Strict convexity is not enough as it may not guarantee the existence of am equilibrium point. For instance $f(x)=e^{-x}$ is strictly convex. So, you will need strict convexity and the existence of a ...
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1 vote
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Low convergence of a perturbation solution to a non-linear ODE

Writing the solution as $$ y(x) = \gamma(\epsilon) (1-x) \ \ \ \quad \mathrm(0) $$ we see that $\gamma$ must satisfy $$ \gamma^3\frac{\epsilon}{40}+\frac{\gamma}{12} -1=0. \ \ \ \quad \mathrm(1) $$ ...
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Choosing a Particular Solution (Undetermined Coefficients)

For a right side term $$p(t)e^{rt}$$ the corresponding solution term is $$t^mq(t)e^{rt}$$ where $p$, $q$ are polynomials with $\deg p=\deg q$ and $m$ is the multiplicity of $r$ as root of the ...
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1 vote

What is the order of accuracy in time and space of the following finite difference scheme?

I took a look back at my notes and found my issue. To obtain the order of accuracy / consistency, we compute the Local Truncation Error: $$R_m^n = \mathcal{L}_{h,k}u_m^n - \mathcal{F}_{h,k}$$ Here, $\...
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1 vote

Why can't a second order differential equation have 2 Neuman conditions?

In elementary terms, whether the solution to the BVP with boundary conditions given in terms of operators $L_a[f]$ and $L_b[f]$ is guaranteed to exist uniquely comes down to whether the matrix $$\...
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1 vote
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How to find the explicit solution of $y'(x)= |y(x)|(1-y(x)) \frac{x^3}{1+x^4}$

The derivative of $(1+x^4)^{1/4}$ is $x^3(1+x^4)^{-3/4}$, which is exactly the integrand you are in doubt of. The equation is separable, so a better substitution would use and preserve that pattern ...
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Is the Sturm-Liouville operator self-adjoint?

$L$ is essentially self-adjoint in the regular case, as you noted. You can show this by exhibiting explicit bounded inverses $(L\pm iI)^{-1}$ on the domain $\mathcal{D}(L)$ consisting of twice ...
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4 votes

Spectrum of the operator on $L^2[0,1]$

An eigenfunction $f$ corresponding to $\lambda\neq 0$ satisfies $f(0)=0.$ Moreover $$f'(x)=-{1\over \lambda}\,f(1-x)$$ Hence $f'(1) =0.$ Applying next derivative gives $$f''(x)=-{1\over \lambda^2}\,f(...
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4 votes
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Quite challenge: how to solve this functional equation?

It's best to transform the equation into one that will differentiate more easily before trying to convert to an ODE. In this particular example, we can convert the integral on the RHS to a second ...
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Quite challenge: how to solve this functional equation?

First rewrite the expression as $$t^{2}w(t)-t^{2}= 2\int_0^{t-a} x \cdot \omega(t - x) dx,$$ and make the change of variable $y=t-x$, to get $$t^{2}w(t)-t^{2}=2\int_a^{t} (t-y)w(y)dy=2t\int_a^{t} w(y)...
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For a linear differential equation with periodic coefficients, unicity implies existence of a solution.

Where did the quoted statement come from? Are you sure you're not leaving something out? For any initial condition, say $X(0) = Y$, there is always exactly one solution to the system with that ...
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1 vote

Parametric solution of the PDE $-yu_x+xu_y=0,\:u(x,x^2)=x^3$

$$-yu_x+xu_y=0$$ The charpit-Lagrange characteristic ODEs are : $$\frac{dx}{-y}=\frac{dy}{x}=\frac{du}{0}$$ A first characteristic equation comes from $du=0$ $$u=c_1$$ which is an obvious particular ...
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3 votes
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Singular solution of differential equation $p^2y+2px-y=0$

Use a solution with a general approach to compare your solution against. Multiply the equation with $y$ to get $$ y^2 = (2py)x+\frac14(2py)^2. $$ This is a Clairaut equation for $v=y^2$, $$ v=v'x+\...
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