Question

Form the differential equation by eliminating arbitrary constants from the relation `y=Ae^(5x)+Be^(-5x)`

Answer 1

`y=Ae^(5x)+Be^(-5x)`

Differentitating w.r.t. x

`dy/dx=A.e^(5x).(5) + Be^(-5x)(-5)`

`therefore dy/dx=5A.e^(5x) - 5Be^(-5x)`

Again differentitating w.r.t. x

`(d^2y)/(dx^2)=5Ae^(5x).(5)-5Be^(-5x).(-5)`

`(d^2y)/(dx^2)=25Ae^(5x)+25Be^(-5x)`

`(d^2y)/(dx^2)=25(Ae^(5x)+Be^(-5x))`

`(d^2y)/dx^2=25y`

`(d^2y)/dx^2-25y=0` is the required differential equation