# `h'(t) = 8t^3 + 5, h(1) = -4` Find the particular solution that satisfies the differential equation.

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### 2 Answers

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.` **

Find the integral

`int(8t^3+5)=h(t)=2t^4+5t+C`

Solve for C using the given point `h(1)=-4`

`2(1)^4+5(1)+C=-4`

C=-11

Thus,

`h(t)=2t^4+5t-11`