Find the family of curves for which the subtangent is…

Find the family of curves for which the subtangent is constant and equal to $a$.

Explanation of Method

Express the subtangent length in terms of $y$ and $y'$, set it equal to the constant $a$, and solve the resulting separable ODE.

Step 1: Set up the ODE

For a curve $y(x)$, the subtangent (the horizontal distance from the point of tangency’s foot to where the tangent meets the $x$-axis) equals $\dfrac{y}{y'}$. Setting this equal to $a$:

\[\frac{y}{y'}=a \quad\Longrightarrow\quad \frac{dy}{y}=\frac{dx}{a},\qquad y\neq0.\]

Step 2: Integrate

\[\ln|y|=\frac{x}{a}+C \quad\Longrightarrow\quad \boxed{\,y=Ce^{x/a}.}\]

Result

Final answer:

\[\boxed{\,y=Ce^{x/a},\qquad C\in\mathbb{R}.}\]