(b) find an expression for ( y = f(x) ) by solving the differential equation ( \frac{dy}{dx} =…

Explanation:

Step1: Separate variables

Separate the variables in the differential equation (\frac{dy}{dx}=\frac{3x^{2}}{y}). We get (y;dy = 3x^{2};dx).

Step2: Integrate both sides

Integrate both sides of the equation. For the left - hand side, (\int y;dy=\frac{y^{2}}{2}+C_{1}). For the right - hand side, (\int3x^{2};dx=x^{3}+C_{2}). So, (\frac{y^{2}}{2}=x^{3}+C) (where (C = C_{2}-C_{1})).

Step3: Use the initial condition

We are given the initial condition (f(2)=-8). Substitute (x = 2) and (y=-8) into the equation (\frac{y^{2}}{2}=x^{3}+C). (\frac{(-8)^{2}}{2}=2^{3}+C). (\frac{64}{2}=8 + C). (32=8 + C), then (C = 24).

Step4: Solve for (y)

From (\frac{y^{2}}{2}=x^{3}+24), we can solve for (y). (y^{2}=2x^{3}+48). Since (y=-8) when (x = 2), (y=-\sqrt{2x^{3}+48}) (we take the negative square root because of the initial condition (y=-8)).

Answer:

(y =-\sqrt{2x^{3}+48})