Question

Form the differential equation of all lines which make intercept 3 on x-axis.

Hint:

Form differential equations by eliminating parameters (e.g., slope, intercept) from the family of lines.

Answer 1

Lines with x-intercept 3 have the form: 
\[ \frac{x}{3} + \frac{y}{b} = 1 \Rightarrow y = -\frac{b}{3} x + b. \] Slope-intercept form: \( y = mx + b \), where x-intercept \( x = 3 \): 
\[ 0 = 3m + b \Rightarrow b = -3m. \] Thus, \( y = mx - 3m \). 
Differentiate: 
\[ \frac{dy}{dx} = m. \] Substitute \( m = \frac{dy}{dx} \) into \( y = mx - 3m \): 
\[ y = \frac{dy}{dx} x - 3 \frac{dy}{dx} \Rightarrow y = x \frac{dy}{dx} - 3 \frac{dy}{dx} \Rightarrow y = \left( x - 3 \right) \frac{dy}{dx}. \] \[ \left( x - 3 \right) \frac{dy}{dx} - y = 0. \] Answer: \( \left( x - 3 \right) \frac{dy}{dx} - y = 0 \).