Question

The differential equation which represents the family of curves y = c₁e^c₂^x, where c₁ and c₂ are arbitrary constants, is

(a) y′′ = y′y 

(b) yy′′ = y′

(c) yy′′ = (y′)² 

(d) y′ = y²

Answer 1

(c) : y = c₁e^c₂^x
differentiating w.r.t. x, we get
y′ = c₁ c₂e^c₂^x = c₂y ...(i)
Again differentiating w.r.t. x
y′′ = c₂y′ ....(ii)
From (i) and (ii) upon division

Which is the desired differential equation of the family of curves.