Solve the differential equation (dP)/dt=4P+a

Solve differential equation ${\displaystyle x\frac{dy}{dx}-2y=x^3{\sin }^2\left(x\right)}$

Please tell how one can identify 1st order
1) Homogeneous differential equations.
2) Homogeneous linear differential equations.
3) Non Homogeneous differential equations.
4) Non Homogeneous linear differential equations.

Write first-order differential equations that express the situation: A student in an engineering course finds the rate of increase of his grade point average GG is directly proportional to the number NN of study hours/week and inversely proportional to the amount AA of time spent on online games .

Solve differential equation ${\displaystyle y^{\prime }+y=x,y\left(0\right)=1}$

What is the derivative of ${\displaystyle h\left(x\right)=\sin 2x\cos 2x}$

Solve differential equation $y^{\prime }+y/x=3y^{-2}$ $x>0$

Classify the equation as separable, linear, both, or neither.
${\displaystyle y\frac{dy}{dx}=2x+y}$