please help me with this differential equation sort of physics application? ?

dA(t)/dt = kA(t).

First you should define

n = n(t) = Number of the remaining atomic nuclei at the time t

dt= infinitesimal small time interval

dn = Number of decayed nuclei in the interval dt

You can assume that dn is proportional to dt and n:

dn~dt

dn~n

and therfore also proportional to the product n*dt:

=> dn~n*dt

With L(=lambda) as proportional factor is

dn = -L*n*dt (1)

The minus before L indicates that the number of nuclei decreases.

(1) is a homogeneous linear DGL of 1. order with const. coefficient L:

dn/dt+L*n=0 or differently written n`+Ln=0

The general solution is n(t)=C* e^-Lt

With the initia value of n(t) you will receive the constant C:

n(0)=n_0 ; (=number of the nuclei to begin of the decay)

=>C=n_0

n(t) = n_0 *e^-Lt

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If A(t) is the amount of substance, the rate of change of A(t) is dA(t)/dt. This is proportional to the amount remaining. If the initial amount is A0, the amount remaining is A0 - A(t), so

dA(t)/dt = k*[A0 - A(t)],