Can anyone help with filling this out What can you say about

Can anyone help with filling this out?

What can you say about the graph of a solution of the equation when x is close to 0? What if x is large? If x is close to 0, then xy3/25 is and hence y\' is Thus, the graph of y must have a tangent line that is nearly If x is large, then xy3/25 is and the graph of y must have a tangent line that is nearly (In both cases, we assume reasonable values for y.) Verify that all members of the family y = 5(c - x2)-1/2 are solutions of the differential equation

Solution

a)When x is close to 0 we have xy³/25 close to 0
Or y\' close to 0. Now for any curve, y\' is the slope of the tangent

Hence when x is close to 0 the graph of y must have a tangent line that\'s nearly horizontal (having slope 0)
Similarly when x is large xy³/25 is large or y\' is large. Hence when x is large the graph of y must have a tangent line that\'s nearly vertical

b) y = 5(c-x²)^-1/2
Differentiating this, we get y\' = 5*(-1/2)*(-2x)(c-x²)^-3/2 =5x*(c-x²)^-3/2
RHS = xy³/25 = x*[ 5(c-x²)^-1/2]³ = 5x*(c-x²)^-3/2 = y\' = LHS
Proved