Recent questions in Mathematics

$Two identical spheres, $X$ and $Y$, are suspended by insulated threads from a ceiling, as shown in the diagram below. Sphere $X$ has an excess of 25 electrons and the charge on $Y$ is unknown.$

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Two identical spheres, $X$ and $Y$, are suspended by insulated threads from a ceiling, as shown in the diagram below. Sphere $X$ has an excess of 25 electrons and the charge on $Y$ is unknown.

asked 4 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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What is the probability of picking a red ball twice from a bag of 4 green balls and 3 red balls, without replacement?

asked 4 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 6 views

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Determine the mode of the set of values: $8,13,20,9,13,15,8,12,13,10,16$ and 11 .

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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$2^{2 x+1}-5\left(2^{x}\right)-12=0$

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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Psychology vs physiology: which leads the person not to quit their addictions?

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Does education have a deep impact on the financial success of an individual?

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 6 views

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A tank originally contains 10 gal of water with $1 / 2 \mathrm{lb}$ of salt in solution.

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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Solve the following IVP. \[ \vec{x}^{\prime}=\left(\begin{array}{cc} -5 & 1 \\ 4 & -2 \end{array}\right) \vec{x}, \quad \vec{x}(0)=\left(\begin{array}{l} 1 \\ 2 \end{array}\right) \]

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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Find the solution to the following system. \[ \begin{aligned} &x_{1}^{\prime}=x_{1}+2 x_{2} \quad x_{1}(0)=0 \\ &x_{2}^{\prime}=3 x_{1}+2 x_{2} \quad x_{2}(0)=-4 \end{aligned} \]

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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Solve the following IVP. \[ \vec{x}^{\prime}=\left(\begin{array}{ll} 1 & 2 \\ 3 & 2 \end{array}\right) \vec{x}, \quad \vec{x}(0)=\left(\begin{array}{c} 0 \\ -4 \end{array}\right) \]

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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What are some of the interesting facts with regards to solutions of systems of differential equations?

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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What is the difference between homogeneous and nonhomogeneous systems of differential equations?

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Convert the following system to matrix form. \[ \begin{aligned} &x_{1}^{\prime}=4 x_{1}+7 x_{2} \\ &x_{2}^{\prime}=-2 x_{1}-5 x_{2} \end{aligned} \]

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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Write the following $4^{\text {th }}$ order differential equation as a system of first order, linear differential equations.

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 12 views

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Write the following $2^{\text {nd }}$ order differential equation as a system of first order, linear differential equations. \[ 2 y^{\prime \prime}-5 y^{\prime}+y=0 \quad y(3)=6 \quad y^{\prime}(3)=-1 \]

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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How do we solve coupled linear ordinary differential equations?

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What is a coupled system of differential equations?

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The coupled system of ordinary differential equations

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Suppose that an elementary magnet is situated at the origin and that its axis corresponds to the $z$-axis.

asked 5 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 6 views

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Consider the system of differential equations

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Consider the Lorenz model of the form \[ \begin{aligned} &\frac{d w}{d t}=R-z y-w \\ &\frac{d z}{d t}=w y-z \\ &\frac{d y}{d t}=\sigma(z-y) \end{aligned} \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Find the solution of the initial value problem \[ \frac{d^{3} u}{d t^{3}}+u=0 \] with $u(t=0)=u_{0}, d u(t=0) / d t=u_{t 0}, d^{2} u(t=0) / d t^{2}=u_{t t 0}$ and $u$ is a real-valued function.

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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A model for epidemics is given by \[ \frac{d S}{d t}=-r S I, \quad \frac{d I}{d t}=r S I-a I, \quad \frac{d R}{d t}=a I \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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Consider the system of differential equations \[ \frac{d P_{k}}{d \tau}=-(k-1) P_{k}-(z k-2 k+2) P_{k+1} \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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The surface of a two-dimensional ellipsoid can be imbedded in three-dimensional Euclidean space by the equation \[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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Within the rotating wave approximation the dynamical system of perturbed Maxwell-Bloch equations is

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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Consider the autonomous system. Show that a gradient system cannot have any limit cycle solutions.

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Consider the coupled Riccati equation \[ \frac{d u_{i}}{d t}=d_{i}+\sum_{j=1}^{N} e_{i j} u_{j}+\sum_{j, k=1}^{N} f_{i j k} u_{j} u_{k}, \quad i=1,2, \ldots, N \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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The Lorenz model is given by \[ \frac{d u_{1}}{d t}=\sigma\left(u_{2}-u_{1}\right), \quad \frac{d u_{2}}{d t}=u_{1} u_{3}+r u_{1}-u_{2}, \quad \frac{d u_{3}}{d t}=u_{1} u_{2}-b u_{3} \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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The Schwarzian derivative plays an important role in several branches of complex analysis. Let $w$ be a holomorphic function. The Schwarzian derivative $\{w ; z\}$ of $w$ is defined by

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 6 views

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The Lorenz model is given by \[ \frac{d x}{d t}=\sigma(y-x), \quad \frac{d y}{d t}=-x z+r x-y, \quad \frac{d z}{d t}=x y-b z \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Show that \[ \frac{d^{3} u}{d x^{3}}-u \frac{d u}{d x}=0 \] admits the first integrals

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 8 views

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Consider the nonlinear ordinary differential equation of third order \[ F \equiv \frac{d^{3} u}{d x^{3}}+4 u \frac{d^{2} u}{d x^{2}}+6 u^{2} \frac{d u}{d x}+3\left(\frac{d u}{d x}\right)^{2}+u^{4}=0 . \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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find the differential equation for $s(t)$, where $\dot{s} \equiv d s / d t$. Here $\kappa_{j}$ are constants.

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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The Halphen-Darboux system of ordinary differential equations is given by

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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Find the first integrals of the system

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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The autonomous system of ordinary differential equations - darboux-halphen system

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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In the study of the Lagrangian structure of the $A B C$-flow we consider the dynamical system

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Consider the initial value problem of the autonomous system

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 9 views

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The geodesic flow on a sphere $S^{n}$ \[ \langle\mathbf{x}, \mathbf{x}\rangle \equiv \mathbf{x}^{T} \mathbf{x}=\|\mathbf{x}\|^{2}=1 \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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Consider the initial value problem for the system of linear first order ordinary differential equations \[ \frac{d \mathbf{u}}{d t}=A \mathbf{u}+\mathbf{b}(t) \]

asked 6 hours ago in Mathematics by ♦Pieter Diamond (40,695 points) | 7 views

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