Question 2.5.1: Solve the nonlinear first-order differential equation

Following our approach in example 2.3.1, we can separate the variables y and t algebraically to arrive at the equation

y^{−2} \frac { dy}{dt}=−1−3t^{2}

Integrating both sides of this equation with respect to t ,

\int {(y(t ))^{−2} \frac { dy}{dt} dt }=\int {(−1−3t^{2}) dt}     (2.5.2)

The left-hand side may be simplified to \int {y^{−2} dy}. Thus, evaluating each integral in (2.5.2), we find that

−y^{−1} =−t −t^{3} +C    (2.5.3)

We note again that since an arbitrary constant of integration arises on each side, it suffices to include just one. It is essential here to observe that by successfully integrating, we have removed the presence of y´ in the equation, and now have only an algebraic, rather than differential, equation in t and y. Solving (2.5.3) algebraically for y, it follows

y = \frac {1}{t + \frac {1}{3}t^{3} −C}