GATE EC Differential Equations questions

The general solution of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x} = \dfrac{1+\cos 2y}{1-\cos 2x}$ is

Consider the differential equation $\dfrac{\mathrm{d} x }{\mathrm{d} t} = 10 – 0.2x$ with initial condition $x(0) = 1$. The response $x(t)$ for $t>0$ is

The state variable representation of a system is given as $\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$ $y=\begin{bmatrix} 0 &1 \end{bmatrix} x$ The response $y(t)$ is

Consider the differential equation $$\dfrac{\mathrm{d^{2}}x(t) }{\mathrm{d} t^{2}} +3\frac{\mathrm{d}x(t)}{\mathrm{d} t} + 2x(t) = 0.$$ Given $x(0) = 20$ and $x(1) = 10/e,$ where $e = 2.718,$ the value of $x(2)$ is ________.

A system is described by the following differential equation, where $u(t)$ is the input to the system and $y(t)$ is the output of the system. $$y(t) + 5y(t) = u(t)$$ When $y(0) = 1$ and $u(t)$ is a unit step function, $y(t)$ is

For the following feedback system $G(s) = \dfrac{1}{(s+1)(s+2)}.$ The $2\%$-settling time of the step response is required to be less than $2$ seconds. Which one of the following compensators $C(s)$ achieves this?

If the characteristic equation of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}+2\alpha \frac{\mathrm{d}y}{\mathrm{d} x}+y= 0$$ has two equal roots, then the value 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 $$\frac{d^2x}{dt^2}+2 \frac{dx}{dt}+x=0$$ is

With initial values $y(0) =y’(0)=1$, the solution of the differential equation $$\frac{d^2y}{dx^2}+4 \frac{dy}{dx}+4y=0$$ at $x=1$ is ________

A system is described by the differential equation $\dfrac{\mathrm{d}^{2} y}{\mathrm{d} x} + 5\dfrac{\mathrm{d}y }{\mathrm{d} x} + 6y(t) = x(t).$ Let $x(t)$ be a rectangular pulse given by $x(t) = \begin{cases} 1&0<t<2 \\ 0&\text{otherwise} \end{cases}$ Assuming that $y(0) = 0$ and $\dfrac{\mathrm{d}y }{\mathrm{d} x} = 0$ at $t=0,$ …

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 $x(1)=0.5$, the solution of the differential equation, $$t\frac{dx}{dt}+x=t$$ is

Consider the differential equation $\frac{d^2y(t)}{dt^2}+2\frac{dy(t)}{dt}+y(t)=\delta(t)$ with $y(t)\big|_{t=0^-}=-2$ and $\frac{dy}{dt}\big|_{t=0^-}=0$. The numerical value of $\frac{dy}{dt}\big|_{t=0^+}$ is

A solution for the differential equation $\qquad \dot{x}(t)+2 x(t)=\delta(t)$ with initial condition $x(0-)=0$ is

For the differential equation $\dfrac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are $y=0$ for $x=0$, and $y=0$ for $x=a$ The form of non-zero solutions of $y$ (where $m$ varies over all integers) are

The following differential equation has $$3 \frac{d^{2} y}{d t^{2}}+4\left(\frac{d y}{d t}\right)^{3}+y^{2}+2=x$$

A solution of the following differential equation is given by $$\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$$

A system described by the following differential equation $\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=x(t)$ is initially at rest. For input $x(t)=2 u(t)$, the output $y(t)$ is

The solution of $X=A(t) X(t)$, is