Question

The solution of the following differential equation represents \[ \frac{dy}{dx} = \frac{y + 1}{x} \]

• A. a straight line
• B. a parabola
• C. an ellipse
• D. a hyperbola

Hint:

When dealing with differential equations of the form \( \frac{dy}{dx} = \frac{f(y)}{g(x)} \), consider using substitution to reduce it to a separable or linear form.

Answer 1

The Correct Option is A

Step 1: Rearranging the equation.
Given: \[ \frac{dy}{dx} = \frac{y + 1}{x} \] This is a first-order linear differential equation. 
Step 2: Substitution to simplify. 
Let \( u = y + 1 \Rightarrow \frac{du}{dx} = \frac{dy}{dx} \). So the equation becomes: \[ \frac{du}{dx} = \frac{u}{x} \] Step 3: Solve the simplified equation. \[ \frac{du}{u} = \frac{dx}{x} \Rightarrow \ln |u| = \ln |x| + C \Rightarrow u = Cx \Rightarrow y + 1 = Cx \Rightarrow y = Cx - 1 \] Step 4: Final form of the solution. The solution is a straight line: \[ y = Cx - 1 \]