Let (X_{1}, X_{2}) be independent random variables. X_{1} has mean 0 and variance 1, while X_{2} has mean 1 and variance 4. The mutual information I (X_{1}; X_{2}) between X_{1} and X_{2} in bits is_________.
Which one of the following is the general solution of the first order differential equation $\frac{dy}{dx}=(x+y-1{)}^{2}$, where x, y are real?
The generation solution of the differential equation $\frac{{d}^{2}y}{d{x}^{2}}+2\frac{dy}{dx}-5y=00$ in terms of arbitrary constant K_{1} and K_{2} is
Which one of the following is a property of the solutions to the Laplace equation: $ \nabla^2f=0? $
The ordinary differential equation
$\frac{dx}{dt}=-3x+2,$ with $ \mathrm x\left(0\right)=1 $
The particular solution of the initial value problem given below is
$ \frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\;\; $ with $y\left(0\right)=3$ and $ \frac{dy}{dx}\vert_{x=0}=-36 $
The solution of the differential equation $\frac{{d}^{2}y}{d{t}^{2}}+2\frac{dy}{dt}+y=0$ with $ y(0)=y'(0)=1 $ is
The general solution of the differential equation $\frac{dy}{dx}=\frac{1+\mathrm{cos}2y}{1-\mathrm{cos}2x}$ is
Consider the differential equation
$\frac{{d}^{2}x\left(t\right)}{d{t}^{2}}+3\frac{dx\left(t\right)}{dt}+2x\left(t\right)=0$
Given x(0) = 20 and x(1) = 10/e, where e = 2.718, the value of x(2) is _____.
If the characteristic equation of the differential equation
$\frac{{d}^{2}y}{d{x}^{2}}+2\alpha \frac{dy}{dx}+y=0$
has two equal roots, then the values of $\alpha $ are
Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively?
If a and b are constants, the most general solution of the differential equation
$\frac{{d}^{2}x}{d{t}^{2}}+2\frac{dx}{dt}+x=0$ is
With initial values y(0)=y′(0)=1, the solution of the differential equation
$\frac{{d}^{2}y}{d{x}^{2}}+4\frac{dy}{dx}+4y=0$
at x=1 is _____.
A system described by a linear, constant coefficient, ordinary, first order differential equation has an exact solution given by y(t) for t > 0 , when the forcing function is x(t) and the initial condition is y(0) . If one wishes to modify the system so that the solution becomes -2y(t) for t > 0 , we need to
With initial condition $x\left(1\right)=0.5$ , the solution of the differential equation,
$t\frac{dx}{dt}+x=t$ is
$\frac{{d}^{2}y\left(t\right)}{d{t}^{2}}+2\frac{dy\left(t\right)}{dt}+y\left(t\right)=\delta \left(t\right)$ with ${\overline{)y\left(t\right)}}_{t={0}^{-}}=-2$ and ${\overline{)\frac{dy}{dt}}}_{t={0}^{-}}=0$
The numerical value of ${\overline{)\frac{dy}{dt}}}_{t={0}^{+}}$ is
The solution of the differential equation $\frac{dy}{dx}=ky,y\left(0\right)=cis$
A function n(x) satisfied the differential equation $\frac{{d}^{2}n\left(x\right)}{d{x}^{2}}-\frac{n\left(x\right)}{{L}^{2}}=0$ where L is a constant. The boundary conditions are: n(0)=K and n ( ∞ ) = 0. The solution to this equation is
Consider differential equation $\frac{dy\left(x\right)}{dx}-y\left(x\right)=x$ with the initial condition y(0) = 0. Using Euler’s first order method with a step size of 0.1, the value of y (0.3) is
The order of the differential equation $\frac{{d}^{2}\mathrm{y}}{d{\mathrm{t}}^{2}}+{\left(\frac{d\mathrm{y}}{d\mathrm{t}}\right)}^{3}+{\mathrm{y}}^{4}={\mathrm{e}}^{-\mathrm{t}}$ is
Match each differential equation in Group I to its family of solution curves from Group II.
Which of the following is a solution to the differential equation $\frac{dx\left(t\right)}{dt}+3x\left(t\right)=0$ ?
The solution of the differential equation ${k}^{2}\frac{{d}^{2}y}{d{x}^{2}}=y-{y}_{2}$ under the boundary conditions (i) y = y_{1} at x = 0 and (ii) y = y_{2} at x = $\infty $, where k, y_{1 }and y_{2} are constants, is