Differential Equations · Engineering Mathematics · GATE ECE

Consider the differential equation $\dot{\vec{w}}=A \vec{w}$, with $\vec{w}(t=0)=\left[\begin{array}{l}1 \\ 1\end{array}\right]$.

If $\vec{w}(t)=e^t \vec{u}_x+e^{-2 t} \vec{u}_y$ be the solution to the equation where $\vec{u}_x$ and $\vec{u}_y$ are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing $A$ ?

The function $y(t)$ satisfies

$$ t^2 y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0 $$

where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is ___________ (rounded off to two decimal places).

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is ____.

Consider the following partial differential equation (PDE)

$$a{{{\partial ^2}f(x,y)} \over {\partial {x^2}}} + b{{{\partial ^2}f(x,y)} \over {\partial {y^2}}} = f(x,y)$$,

where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters $$\xi $$ and $$\eta $$ such that $$f(x,y) = {e^{\xi x + \eta y}}$$ is a solution of the given PDE.

The general solution of $\frac{d^2 y}{d x^2}-6 \frac{d y}{d x}+9 y=0$ is

$${\rm I}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly independent on $$ - 1 \le x \le 0$$
$${\rm II}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly dependent on $$0 \le x \le 1$$
$${\rm III}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly independent on $$0 \le x \le 1$$
$${\rm IV}.\,\,\,\,\,$$ $${y_1},{y_2}$$ and $${y_3}$$ are linearly dependent on $$ - 1 \le x \le 0$$

Which one among the following is correct?

Which one of the following options contains two solutions of the differential equation $\frac{d y}{d x}=(y-1) x$ ?

The value of $$x(2)$$ is

Group $$I$$
$$P:$$$$\,\,\,$$ $${{dy} \over {dx}} = {y \over x}$$
$$Q:$$$$\,\,\,$$ $${{dy} \over {dx}} = {{ - y} \over x}$$
$$R:$$$$\,\,\,$$ $${{dy} \over {dx}} = {x \over y}$$
$$S:$$$$\,\,\,$$ $${{dy} \over {dx}} = {{ - x} \over y}$$

Group $$II$$
$$(1)$$$$\,\,\,$$ Circle
$$(2)$$$$\,\,\,$$ straight lines
$$(3)$$$$\,\,\,$$ Hyperbola

List-$${\rm I}$$
$$(P)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}y{{dy} \over {dx}} + {a_3}y = {a_4}$$
$$(Q)$$ $${a_1}{{{d^2}y} \over {d{x^3}}} + {a_2}y = {a_3}$$
$$(Q)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}x{{dy} \over {dx}} + {a_3}{x^2}y = 0$$

List-$${\rm II}$$
$$(1)$$ Non-linear differential equation
$$(2)$$ Linear differential equation with constants coefficients
$$(3)$$ Linear homogeneous differential equation
$$(4)$$ Non-linear homogeneous differential equation
$$(5)$$ Non-linear first order differential equation