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

`(da)/(dt) =-9a`

This ODE (Ordinary Differential Equation) is separable.
That is,

`(da)/(a) = -9`

Integrate to obtain

`\int (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.