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

`y=3t^9+C`

Explanation:

Given:
`(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$\\y=(27t^(8+1))/(8+1) +C$, (where C is the constant of integration)$\\y=(27t^(9))/(9) +C\\\\y=3t^9+C`

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

CHECK:

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