(d) Solve the initial value problem (find $y$ in terms of $x$ ) if $y^{\prime}(x)=8 x^{3}-12 x^{-7}, y(1)=4$

Step 1 ：Integrate the given derivative y'(x) with respect to x: \(\int (8x^3 - 12x^{-7}) dx\)

Step 2 ：Find the general solution for y(x): \(y(x) = 2x^4 + 2x^{-6} + C\)

Step 3 ：Apply the initial condition y(1) = 4: \(4 = 2(1)^4 + 2(1)^{-6} + C\)

Step 4 ：Solve for the constant of integration C: \(C = 0\)

Step 5 ：Write the final solution for y(x): \(\boxed{y(x) = 2x^4 + 2x^{-6}}\)