maths: differencial equations?

Question

The rate at which a radioactive specimen decays is given by

d𝑁/d𝑡=−𝛼𝑁

where 𝑁 is the number of radioactive particles at time 𝑡 and 𝛼 is a constant. Solve this equation to find an expression for 𝑁 as a function of time given that 𝑁=𝑁0 at 𝑡=0. Hence find the half-life of the specimen i.e. the time taken for the number of radioactive particles to halve.

Answer 1

dN/N = -alpha dt =>

N = c e^(-alpha t),

and N(0) = No implies

N = No * e^(-alpha t).

The half-life is the time when

1/2 = e^(-alpha t), or

ln(2) = alpha t, or

t = [ln(2)]/alpha.

Answer 2

The answer is as follows: