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The solution of the initial value problem:

$$dy/dx = -9x^2$$

$$y(0) = 4$$

what is $y$ ?

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2 Answers

up vote 2 down vote accepted

$dy=-9x^2 dx \Rightarrow y=-9\cdot\int x^2 \,dx \Rightarrow y=-9 \cdot \frac{x^3}{3} +C \Rightarrow$

$\Rightarrow y=-3x^3+C \Rightarrow 4=-3 \cdot 0 +C \Rightarrow C=4 \Rightarrow$

$y=-3\cdot x^3+4$

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Hints:

The derivative of $y$ is $-9x^2$. The antiderivative of $-9x^2$ is therefore ...

The antiderivative of $-9x^2$ will include an additive constant $C$. Use the initial condition $y(0)=4$ to find the value of $C$.

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