Question

If show that

Answer 1

Given, y=A e mx +B e nx .

The first order derivative is obtained by differentiating the function with respect to x.

dy dx = d dx ( A e mx +B e nx ) dy dx =A e mx d dx ( mx )+B e nx d dx ( nx ) =Am e mx +Bn e nx

Again differentiate the above function with respect to x.

d 2 y d x 2 = d dx ( Am e mx +Bn e nx ) =A m 2 e mx +B n 2 e nx

Substitute the value of first order and second order derivative in the equation given below.

d 2 y d x 2 −( m+n ) dy dx +mny=A m 2 e mx +B n 2 e nx −( m+n )( Am e mx +Bn e nx ) +mn( A e mx +B e nx ) =A m 2 e mx +B n 2 e nx −A m 2 e mx −Bmn e nx −Amn e mx −B n 2 e nx +Amn e mx +Bmn e nx =0

Hence, it is proved that d 2 y d x 2 −( m+n ) dy dx +mny=0.