Question

The differential equation \[ \frac{dy}{dx}=-\left(\frac{x+y}{1+x^2}\right) \] is

• A. Of variable separable form
• B. First order linear equation
• C. Homogeneous
• D. Exact differential equation

Hint:

A first order linear differential equation has the form \(\frac{dy}{dx}+P(x)y=Q(x)\).

Answer 1

The Correct Option is B

Step 1: Given: \[ \frac{dy}{dx}=-\frac{x+y}{1+x^2} \]

Step 2: Split: \[ \frac{dy}{dx}=-\frac{x}{1+x^2}-\frac{y}{1+x^2} \]

Step 3: Bring \(y\)-term to left: \[ \frac{dy}{dx}+\frac{1}{1+x^2}y=-\frac{x}{1+x^2} \]

Step 4: This is of the form: \[ \frac{dy}{dx}+P(x)y=Q(x) \]

Step 5: Hence, it is a first order linear equation. \[ \boxed{\text{First order linear equation}} \]