Chapter 1 - Introduction - 1.3 Classification of Differential Equations - Problems - Page 25: 9

$t(\frac{d}{dt}(3t+t^{2}))-(3t+t^{2}) = t^{2}$.

We must verify that $y = 3t+t^{2}$ is a solution to the differential equation $ty'-y = y^{2}$. This merely requires substituting the proposed solution for the variable $y$ in the equation and performing the indicated operations. This means simplifying $t(\frac{d}{dt}(3t+t^{2}))-(3t+t^{2})$. Thus, $(3+2t)-(3t+t^{2})$ $=3t+2t^{2}-3t-t^{2}$ $=t^{2}$, which was to be shown.