The solution of the initial value problem:
$dy/dx = -9x^2$
$y(0) = 4$
what is $y$ ?
The solution of the initial value problem:
$dy/dx = -9x^2$
$y(0) = 4$
what is $y$ ?
$dy=-9x^2 dx \Rightarrow y=-9\cdot\int x^2 \,dx \Rightarrow y=-9 \cdot \frac{x^3}{3} +C \Rightarrow$
$\Rightarrow y=-3x^3+C \Rightarrow 4=-3 \cdot 0 +C \Rightarrow C=4 \Rightarrow$
$y=-3\cdot x^3+4$
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 $C$.