Answer:
[tex]y=3t^9+C[/tex]
Step-by-step explanation:
Given: [tex]\dfrac{dy}{dt}=27t^8[/tex]
We want to obtain the general solution of the given differential equation.
[tex]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[/tex]
The general solution of the differential equation is: [tex]y=3t^9+C[/tex]
CHECK:
[tex]\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[/tex]
