The "order" of the following ordinary differential equation is $\qquad$ .

$$ \frac{d^3 y}{d x^3}+\left(\frac{d^2 y}{d x^2}\right)^6+\left(\frac{d y}{d x}\right)^4+y=0 $$

A partial differential equation

$$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$

is defined for the two-dimensional field $T: T(x, y)$, inside a planar square domain of size 2 m × 2 m. Three boundary edges of the square domain are maintained at value $T = 50$, whereas the fourth boundary edge is maintained at $T = 100$.

The value of $T$ at the center of the domain is

Consider two Ordinary Differential Equations (ODEs):

P: $ \dfrac{dy}{dx} = \dfrac{x^4 + 3x^2 y^2 + 2y^4}{x^3 y} $

Q: $ \dfrac{dy}{dx} = -\dfrac{y^2}{x^2} $

Which one of the following options is CORRECT?

The second-order differential equation in an unknown function $$u : u(x, y)$$ is defined as $$\frac{\partial^2 u}{\partial x^2}= 2$$

Assuming $$g : g(x)$$, $$f : f(y)$$, and $$h : h(y)$$, the general solution of the above differential equation is

For the following partial differential equation,

$x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2}$

which of the following option(s) is/are CORRECT?

The function f(x, y) satisfies the Laplace equation

$$\Delta$$²f(x, y) = 0

on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. The value of this function on the circular boundary of this domain is equal to 3. The numerical value of f/(0, 0) is :

Consider the following expression:

z = sin(y + it) + cos(y $$-$$ it)

where z, y, and t are variables, and $$i = \sqrt { - 1} $$ is a complex number. The partial differential equation derived from the above expression is

For the equation

$${{{d^3}y} \over {d{x^3}}} + x{\left( {{{dy} \over {dx}}} \right)^{3/2}} + {x^2}y = 0$$

the correct description is

Let $y$ be the solution of the initial value problem $y^{\prime}+0.8 y+0.16 y=0$ where $y(0)=3$ and $y^{\prime}(0)=4.5$. Then, $y(1)$ is equal to__________ (rounded off to 1 decimal place).

A 2 m × 2 m tank of 3 m height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water. At $ t = 0 $, an inflow of a salt solution of concentration 5 g/ $ m^3 $ at the rate of 2 litre/s and an outflow of the well stirred mixture at the rate of 1 litre/s are initiated. This process can be modelled using the following differential equation:

$$ \frac{dm}{dt} + \frac{m}{6000 + t} = 0.01 $$

where $ m $ is the mass (grams) of the salt at time $ t $ (seconds). The mass of the salt (in grams) in the tank at 75% of its capacity is ______________ (rounded off to 2 decimal places).

The solution of the differential equation

$\rm \frac{d^3y}{dx^3}-5.5\frac{d^2y}{dx^2}+9.5\frac{dy}{dx}-5y=0$

is expressed as 𝑦 = 𝐶₁𝑒^2.5𝑥 + 𝐶₂𝑒^𝛼𝑥 + 𝐶₃𝑒^𝛽𝑥 , where 𝐶₁, 𝐶₂, 𝐶₃, 𝛼, and 𝛽 are constants, with α and β being distinct and not equal to 2.5. Which of the following options is correct for the values of 𝛼 and 𝛽?

The differential equation,

$\rm \frac{du}{dt}+2tu^2=1,$

is solved by employing a backward difference scheme within the finite difference framework. The value of 𝑢 at the (𝑛 − 1) th time-step, for some 𝑛, is 1.75. The corresponding time (t) is 3.14 s. Each time step is 0.01 s long. Then, the value of (un - un -1) _________ is (round off to three decimal places). 

Consider the differential equation

$${{dy} \over {dx}} = 4(x + 2) - y$$

For the initial condition y = 3 at x = 1, the value of y at x = 1.4 obtained using Euler's method with a step-size of 0.2 is ________. (round off to one decimal place)

Which of the following is the solution of the above equation ($$C$$ is an arbitrary constant)