`h'(t) = 8t^3 + 5, h(1) = -4` Find the particular solution that satisfies the differential equation.
You need to use direct integration to evaluate the general solution to the differential equation:
`int (8t^3 + 5)dt = int 8t^3 dt + int 5dt`
`int (8t^3 + 5)dt = 8t^4/4 + 5t + c`
`int (8t^3 + 5)dt = 2t^4 + 5t + c`
You need to find the particular solution using the information provided by the problem, that h(1) = -4, such that:
`h(1) =2*1^4 + 5*1 + c => -4 = 7 + c => c = -4 - 7 => c = -11`
Hence, evaluating the particular solution to the given differential equation yields `h(t) = 2t^4 + 5t - 11.`