Answer to Question #124115 in Calculus for Tilly Nill

Question

Verify that the given family of functions solves the differential equation.
(i) dy⁄dx= (1-2t)y², y=⅟(C – t + t²)
(ii) dy⁄dx= y² sin t, y=⅟(C + cos t)

Answer 1

$(i) \frac{dy}{dt}=(1-2t)y^2\\[1 em] \text{We separate y and t to get}\\[1 em] \frac{dy}{y^2}=(1-2t)dt\\[1 em] \text{Integrate}\\[1 em] \int \frac{dy}{y^2}=\int(1-2t)dt\\[1 em] \therefore \frac{-1}{y}=(t-t^2+c)\\[1 em] \therefore y= \frac{-1}{t-t^2+c}=\frac{1}{C-t+t^2}\\[1 em] (ii) \frac{dy}{dt}=y^2 \sin t\\[1 em] \text{We separate y and t to get}\\[1 em] \frac{dy}{y^2}=\sin t ~dt \\[1 em] \text{Integrate}\\[1 em] \int \frac{dy}{y^2}=\sin t ~dt \\[1 em] \therefore \frac{-1}{y}=-\ cos~ t +c\\[1 em] \therefore y= \frac{-1}{-\ cos ~t +c}=\frac{1}{\ cos ~t +~C}\\[1 em]$