# Tag Info

### 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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### 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 $b$: $$A = \sqrt{a^2+b^2}$$ $$\phi = \arccos(\frac{a}{A})=\arcsin(\frac{b}{A})$$
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
Accepted

### 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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Accepted

### 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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Accepted

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### 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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### 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
Accepted

### 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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1 vote
Accepted

### 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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Accepted

### 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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### 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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### 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+\...