Question

Find the general solution of the differential equation and check the result by differentiation. (Use C for the constant of integration.) dy dt = 27t8

Answer 1

Answer:

$y=3t^9+C$

Step-by-step explanation:

Given:  $\dfrac{dy}{dt}=27t^8$

We want to obtain the general solution of the given differential equation.

$dy=27t^8 dt\\ Take the integral of both sides\\\int dy =\int 27t^8 dt \\y=\dfrac{27t^{8+1}}{8+1} +C , (where C is the constant of integration) \\y=\dfrac{27t^{9}}{9} +C\\\\y=3t^9+C$

The general solution of the differential equation is: $y=3t^9+C$

CHECK:

$\dfrac{d}{dt} y=\dfrac{d}{dt}(3t^9+C)=\dfrac{d}{dt}(3t^9)+\dfrac{d}{dt}(C)\\\\\text{Since derivative of a constant is zero}\\\\\dfrac{dy}{dt}=27t^{9-1}\\\dfrac{dy}{dt}=27t^8$