Which method is appropriate for solving the first‑order differential equation $\frac{d y}{d x} + y = y^{2} e^{x}$ ?

medium MCQ 2 marks Class 12 · Mathematics · Differential Equations

Answer

Bernoulli’s method

The correct answer is Bernoulli’s method — Thus the Bernoulli substitution is the required method.

Step-by-step solution

Rewrite the equation as $\frac{d y}{d x} + y = e^{x} y^{2}$ .
Observe that it is of the Bernoulli form $\frac{d y}{d x} + P (x) y = Q (x) y^{n}$ with $n = 2$ .
Divide both sides by $y^{2}$ to obtain $y^{- 2} \frac{d y}{d x} + y^{- 1} = e^{x}$ .
Set $v = y^{- 1}$ , then $\frac{d v}{d x} = - y^{- 2} \frac{d y}{d x}$ .
Substitute to get a linear equation in $v$ : $\frac{d v}{d x} - v = - e^{x}$ .
Solve the linear equation using an integrating factor $e^{- x}$ .
Finally revert $v = y^{- 1}$ to obtain $y$ .
Thus the Bernoulli substitution is the required method.

Common mistake: Treating the equation as linear without the substitution

v = y^{1 - n}

, which leads to an incorrect integrating factor.

Concept tested: Bernoulli differential equation