Question

Obtain the differential equation by eliminating the arbitrary constants from y = c₁ cos (log x) + c₂ sin (log x).

Answer 1

y = c₁ cos (log x) + c₂ sin (log x)   ...(1)

Differentiating w.r.t. x, we get

`dy/dx = -c_1 sin (log x) * d/dx (log x) + c_2 cos (log x) * d/dx (log x)`

= `(-c_1 sin (log x))/x + (c_2 cos (log x))/x`

∴ `x dy/dx = -c_1 sin (log x) + c_2 cos (log x)`

Differentiating again w.r.t. x, we get

`x (d^2y)/(dx^2) + dy/dx xx 1 = (-c_1 cos (log x))/x - (c_2 sin (log x))/x`

∴ `x^2 (d^2y)/(dx^2) + x dy/dx = - [c_1 cos (log x) + c_2 sin (log x)]`

= – y   ...[By (1)]

∴ `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`

This is the required D.E.