Since the most basic and common computation in science involves natural exponential decay and growth, probably, the most commonly evaluated differential equation is the evaluation of ODE to obtain the natural exponential growth or decay, y(t), . That is, the evaluation of
dy/dt = -ky, where k is constant and has initial condition;
y(t1) = b.
To solve this, simply use the method of separation of variable to rewrite the equation into;
dy/y = -k dt
Now, integrate to obtain;
ln|y| = -kt + c, where c is constant.
Simplify to get;
y(t) = Cexp(-kt), where C is constant.
Using the initial condition;
C= b/exp(-kt1).
Therefore;
y(t) = exp(-kt)*(b/exp(-kt1)) .