Question

Solve the differential equation. question is pictured with answer choices

Answer 1

Integrate both sides with respect to t :

∫ dy/dt dt = ∫ -12t ² dt

y(t) = -4t ³ + C

Use the initial condition to solve for C :

5 = -4•0³+ C

C = 5

So

y(t) = -4t ³ + 5

and the answer is D.

Alternatively, you can directly apply the fundamental theorem of calculus:

$\dfrac{\mathrm dy}{\mathrm dt}=-12t^2\implies y(t)=y(0)+\displaystyle\int_0^t -12u^2\,\mathrm du$

$\implies y(t)=5-4u^3\bigg|_0^t$

$\implies y(t)=5-4t^3$