What is the solution for (dh)/(dt)=4-16t^2,h(1)=0 ?

The solution for (dh)/(dt)=4-16t^2,h(1)=0 is h=4t-(16t^3)/3+4/3

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(dh)/(dt)=4-16t^2,h(1)=0

\frac{d h}{d t} = 4 - 16 t^{2}, h (1) = 0

h = 4 t - \frac{16 t ^{3}}{3} + \frac{4}{3}

Solution steps

\frac{d h}{d t} = 4 - 16 t^{2}

Substitute

\frac{d h}{d t}

with

h^{'} (t)

h^{^{'}} (t) = 4 - 16 t^{2}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

h = \int 4 - 16 t^{2} d t

\int 4 - 16 t^{2} d t = 4 t - \frac{16 t ^{3}}{3} + c_{1}

h = 4 t - \frac{16 t ^{3}}{3} + c_{1}

Apply initial conditions:

h = 4 t - \frac{16 t ^{3}}{3} + \frac{4}{3}

h = 4 t - \frac{16 t ^{3}}{3} + \frac{4}{3}

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\int (y^{2} - 1) d y

x^{3} - 13 x^{2} + 40 x, - x^{3} + 13 x^{2} - 40 x

\int 900 - 36 x^{2} d x

What is the solution for (dh)/(dt)=4-16t^2,h(1)=0 ?
The solution for (dh)/(dt)=4-16t^2,h(1)=0 is h=4t-(16t^3)/3+4/3