If the dependent variable `y` is changed to `z` by the substitution `y=tanz` and the differential equation `(d^2y)/(dx^2)=1+(2(1+y))/(1+y^2)((dy)/(dx))^2` is changed to `(d^2z)/(dx^2)=cos^2z+k((dz)/(dx))^2,` then the value of `k` equal to_______

Answer

2

Explanation

Given, `y=tanz`
`therefore (dy)/(dx)=sec^(2)z.(dz)/(dx)`………..(1)
Now, `(d^(2)y)/(dx^(2))=sec^(2)z.(d^(2)z)/(dx).d/(dx)(sec^(2)z)` [using product rule]
`=sec^(2)z. (d^(2)z)/(dx^(2))+(dz)/(dx).(d)/(dz)(sec^(2)z)(dz)/(dx)`
`(d^(2)y)/(dx^(2))=sec^(2)z.((dz)/(dx))^(2).2sec^(2)z.tanz`...........(2)
Now, `1+(2(1+y))/(1+y^(2))((dy)/(dx))^(2)`
=`1+(2(1+tanz))/(sec^(2)z).sec^(4)z.((dz)/(dx))^(2)`
`=1+2(1+tanz).sec^(2)z.((dz)/(dx))^(2)`
`=1+2sec^(2)z((dz)/(dx))^(2)+2tanz.sec^(2)z((dz)/(dx))^(2)`........(3)
From (2) and (3), we have RHS or (2) = RHS of (3)
or `sec^(2)z.(d^(2)z)/(dx^(2))=1+2sec^(2)z((dz)/(dx))^(2)`
or `(d^(2)z)/(dx^(2))=cos^(2)+2((dz)/(dx))^(2)`
or `k=2`