Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)
Question Statement:
Bernoulli's non-linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)y^n$ ($n \ne 0, 1$) is converted into a linear first-order ODE by the substitution:
Options:
(A) $v = y^{1-n}$
(B) $v = y^n$
(C) $v = y^{n-1}$
(D) $v = y^{-n}$
Answer Key:
A
Step-by-Step Solution & Explanation
Dividing Bernoulli's equation by $y^n$:
$y^{-n} \frac{dy}{dx} + P(x) y^{1-n} = Q(x)$
Let $v = y^{1-n}$. Then $\frac{dv}{dx} = (1-n) y^{-n} \frac{dy}{dx} \implies y^{-n} \frac{dy}{dx} = \frac{1}{1-n} \frac{dv}{dx}$.
Substituting yields the linear ODE:
$\frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x)$
Hence, option A is correct.