This paper shows in a simple and direct way the equivalence of the nonlinear differential equation $y'' + r(x)y' + q(x)Z(y) = A(y)y'^2 + g(x)z(y)[u(y)]^a $, $Z(y) = z(y)u(y)$, to the linear equation $L_1 u = g(x)$, $a = 0$, or to the nonlinear equation $L_1 u = g(x)u^a $, $a \ne 0$, where $L_1 = {{d^2 } / {dx^2 }} + r(x){d / {dx}} + q(x)$. The two differential equations in which $A(y)$ is equal to $y^{ - 1} $ or to $(1 - l)y^{ - 1} $ serve as particular examples. Some nonlinear equations in u are solvable for certain values of the exponent a. An analogous class of nonlinear partial differential equations is presented. These results generalize the earlier work of Herbst.