Question

Find the general form of the function that satisfies da/dt = -9a

Answer 1

Da/dt=-9a
t is variable. integral of -9adt= -9a*integral dt= -9a*t+c
The general form of funcion that satisfies differential equations da/dt=-9a is a=-9a*t+c

A solution of differential equation is a function which satisfies the equation.

Answer 2

The given equation is
$\frac{da}{dt} =-9a$

This ODE (Ordinary Differential Equation) is separable.
That is,
$\frac{da}{a} = -9$

Integrate to obtain
$\int \frac{da}{a} =-9 \int dt \\\\ ln(a) = -9t + k \\\\ a(t) = c \, e^{-9t}$
where k, c are constants.

Answer:  $a(t) = c \, e^{-9t}$, c = constant.