For what value of k, does the differntial equation `(dy)/(dx)=ky` represent the law of natural decay ?

Options

A. -5
B. 0
C. 0.01
D. `(10)^(-1)`

Answer

A

Explanation

` (dy)/(dx) = ky or (dy)/y = kdx`
Integrating both the sides
` int (dy)/y = k int (dx) +log c`
`log y =kx =log c Rightarrow log y - log c = kx `
`or, log (y/c) = kx `
`or, y/c = e^(kx)`
` or , y =c .e^(kx)`
The equation will show a decay. if value of k is negative. only option (a) shows negative value of k.