Question

Bernoulli's equation is of the form

• A. $\left( \frac{dy}{dx} \right) + y = Qy$
• B. $\left( \frac{dy}{dx} \right)^2 + y^n = Qy$
• C. $\left( \frac{dy}{dx} \right) + Py = Qy^n$
• D. $\left( \frac{d^2y}{dx^2} \right) + Py = Qy^n$

Hint:

Remember that Bernoulli's equation is "almost" linear. The presence of the $y^n$ term on the right-hand side is the only thing preventing it from being a standard first-order linear equation. The substitution $v = y^{1-n}$ is the "magic key" that removes this non-linearity.

Answer 1

The Correct Option is C

1. Standard Form: The general form of Bernoulli's equation is: $$\frac{dy}{dx} + P(x)y = Q(x)y^n$$ Where $P(x)$ and $Q(x)$ are continuous functions of $x$, and $n$ is a real number.

2. Special Cases:

• If $n=0$, the equation becomes a standard first-order linear differential equation: $\frac{dy}{dx} + Py = Q$.

• If $n=1$, the equation is also linear and separable: $\frac{dy}{dx} + (P-Q)y = 0$.

3. Method of Solution: To solve the equation when $n \neq 0$ and $n \neq 1$, we divide the entire equation by $y^n$: $$y^{-n} \frac{dy}{dx} + Py^{1-n} = Q$$ Then, we apply the substitution $v = y^{1-n}$. Differentiating $v$ with respect to $x$ yields: $$\frac{dv}{dx} = (1-n)y^{-n} \frac{dy}{dx}$$ Substituting these into the equation transforms it into a linear equation in terms of $v$ and $x$, which can then be solved using an integrating factor. Therefore, Option (C) correctly represents the required mathematical structure.