An equation containing the derivative of a function is called a differential equation.
Depending on the order of the derivatives it is of the first, second or higher order.
The simplest differential equation is an ordinary linear homogenous differential equation of the first order:
y’ + 3y = 0.
The solution to this equation is given with the integrating factor: e-3x
Where the exponent is the primitive function for the coefficient in front of y.
Multiplication of both sides with this factor gives:
e-3x y’ – 3y e-3x = 0
The left side is identical to the derivative of the product D(y e-3x ).
Therefore integration of both sides generates y e3x =C and y = C e-3x
which is the solution C being an arbitrary constant.
For inhomogenous equations the solution is given by the sum of the solution to the homogenous equation and a particular solution: y = yp + yh.
The particular solution is found by proposing a soultion of the same kind as the type of function standing to the right of the equal sign.