The substitution \(y = vx\) reduces the differential equation \(\displaystyle\frac{dy}{dx} = \frac{y}{x} + \sec\left(\frac{y}{x}\right)\) to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(x\displaystyle\frac{dv}{dx} = \sec v\)
The right-hand side depends only on \(\frac{y}{x}\), so the equation is homogeneous. With \(y = vx\), the product rule gives
\( \displaystyle \frac{dy}{dx} = v + x\frac{dv}{dx} \)
and \(\frac{y}{x} = v\). Substituting:
\( \displaystyle v + x\frac{dv}{dx} = v + \sec v \)
\( \displaystyle x\frac{dv}{dx} = \sec v \)
Why the other options are incorrect
Option (2) \(x\displaystyle\frac{dv}{dx} = v + \sec v\) — Incorrect. The v on the left, from \(\frac{dy}{dx} = v + x\frac{dv}{dx}\), is not cancelled with the v on the right.
Option (3) \(\displaystyle\frac{dv}{dx} = \sec v\) — Incorrect. This treats \(\frac{dy}{dx}\) as \(\frac{dv}{dx}\) and drops the factor x from the product rule.
Option (4) \(x\displaystyle\frac{dv}{dx} = v\sec v\) — Incorrect. The two terms on the right are added, not multiplied; after cancelling v only \(\sec v\) is left.