What is the second derivative and the particular solution of this problem? (AP Calculus)

Question

If dy/dx=x/y²:

-Find d²y/dx² (in terms of x & y)

-Find the particular solution of the given differential equation that satisfies the initial condition y(0)=2.

Answer 1

rearrange the equation to get y²dy = xdx and then integrate both sides to get (1/3)*y³ = .5*x²+C.

y = (1.5*x²+C)^1/3 and plug in 0 for x, you know y should be 2 so find the C value that makes it 2.

for 2nd derivative just use quotient rule and implicit differentiation: y'' = (y²-x*2y*y')/y⁴