Question

If \( \frac{a}{a_1} = \frac{b}{b_1} \), then the substitution used to solve the differential equation \(\frac{dy}{dx} = \frac{ax + by + c}{a_1x + b_1y + c_1} \) using separation of variables is:

• A. \( x = x + h, y = y + k \)
• B. \( ax + by = Z \)
• C. \( y = V(x) \cdot x \)
• D. \( x = at, y = bt \)

Hint:

For first-order differential equations with linear numerators and denominators, check for constant ratios in coefficients. If the ratio \( \frac{a}{a_1} = \frac{b}{b_1} \), use substitution of the form \( ax + by = z \) to reduce the equation.

Answer 1

The Correct Option is B

Given that the coefficients are in proportion: \[ \frac{a}{a_1} = \frac{b}{b_1} \Rightarrow \text{ the lines are parallel and the equation is homogeneous in form.} \] Let: \[ z = ax + by \Rightarrow \frac{dz}{dx} = a + b \frac{dy}{dx} \] Substitution \( z = ax + by \) converts the differential equation to a separable form.