d2dx2 y 0 Put your solution in form of sin nx cos nxSolut

d^2/dx^2 + y = 0 Put your solution in form of sin nx & cos nx

Solution

Given that

( d²y / dx² ) + y = 0

D- operator form is,

( D² + 1 ) y = 0

The auxialary equation is ,

m2 + 1 = 0

m² = -1

m = -1

m = ± i [since, i² = -1 ]

m = 0 ± i [ since, it is in the form of ± i ]

= 0 ,   = 1

Hence,

The roots are imaginary.

We know that,

If the roots are imaginary then the solution is ,

y = e^x [c₁cos(x) + c₂ sin(x) ]

    y = e^0.x [c₁cos(1.x) + c₂ sin(1.x) ]

y =  e⁰ [c₁cos(x) + c₂ sin(x) ]

y = c₁ cos(x) + c₂ sin(x) [ since , e⁰ = 1 ]

Therefore,

The general solution is ,    y = c₁ cos(x) + c₂ sin(x)