Question

A first order differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$, where $n$ is a real number is called}

• A. Lagrange Equation
• B. Bernoulli Equation
• C. Laplace Equation
• D. Wave Equation

Hint:

Recognize the presence of $y^n$ on the right-hand side as the defining characteristic of a Bernoulli differential equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Certain non-linear first-order differential equations can be transformed into linear equations using appropriate substitutions.

Step 2: Detailed Explanation:
The given equation is:
\[ \frac{dy}{dx} + P(x)y = Q(x)y^n \]
This is the standard form of the Bernoulli differential equation.
- If $n = 0$ or $n = 1$, the equation is already linear.
- For any other real number $n \neq 0, 1$, the equation is non-linear but can be linearized by dividing both sides by $y^n$ and substituting $u = y^{1-n}$.
Therefore, this equation is called the Bernoulli Equation.

Step 3: Final Answer
The correct option is (B).