Q$1.$

In thermodynamic systems, nonlinear differential equations arise when modeling heat exchange in nonlinear materials.

Verify whether the following function satisfies the equation:

$y^{\text{'}}=12+y^3$

Given:

$y=tan(x)+x^2$

Does this function satisfy the differential equation?

Q$2.$

In studying the decay of a star's brightness over time, scientists often use power functions.

Suppose:

$y=x^m$

Find the values of mmm that satisfy the differential equation:

$x^2{y}^{\text{'}\text{'}}+3x{y}^{\text{'}}=0$

This models the fading light intensity of a distant star.

Q$3.$

The vertical oscillation of a drone's rotor due to a gust of wind can be modeled by the secondorder differential equation:

$x^{\text{'}\text{'}}+4x=0$

Given the solution:

$x(t)=c_{1}cos(2t)+c_{2}sin(2t)$

Solve the initial value problem:

$x(0)=1,x^{\text{'}}(0)=-2$

This represents the vibration of the rotor at the initial moment after impact.

Q$4.$

The growth of a species population $N(t)$ with environmental influence can be modeled as:

$\frac{dN}{dt}+2N=t^{2}N{e}^t$

Use Separation of Variables to solve the equation and determine how the population evolves over time. please solve