A 2 m Γ 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 π¦ = πΆ1π2.5π₯ + πΆ2ππΌπ₯ + πΆ3ππ½π₯ , where πΆ1, πΆ2, πΆ3, πΌ, and π½ are constants, with Ξ± and Ξ² being distinct and not equal to 2.5. Which of the following options is correct for the values of πΌ and π½?
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 π’ at the (π β 1) th time-step, for some π, 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).