Answer to Question #124045 in Calculus for Nii Laryea

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

1) Given $y=\frac{1}{C – t + t²}$ ________(1)

$\implies \frac{dy}{dt} = (-1)(C-t+t^2)^{-2} (-1+2t) = \frac{1-2t}{(C-t+t^2)^2} = (1-2t) y^2$

Hence, $y$ given by (1) is solution of $\frac{dy}{dx} = (1-2t)y^2$ .

2) Given $y=\frac{1}{C + cos t}$ __________(2)

$\implies \frac{dy}{dt} = (-1)(C+cost)^{-2} (-sint) = y^2 sin(t)$

Hence, y given by (2) is solution of $\frac{dy}{dt} = y^2sin(t)$