What can you say about the graph of a solution of the equation y' = $\frac{x y^{3}}{36}$ $(xy^3)/36$ when x is close to 0?

Question

(a) What can you say about the graph of a solution of the equation y' = $\frac{x y^{3}}{36}$ $(xy^3)/36$ when x is close to 0? What if x is large?

If x is close to 0, then y' = $\frac{x y^{3}}{36}$ $(xy^3)/36$ is (choices are: large, close to 0, and close to 36) , and hence y' is (choices are: large, close to 0, and close to 36) Thus, the graph of y must have a tangent line that is nearly (choices are: horizontal, vertical). If x is large, then $\frac{x y^{3}}{36}$ $(xy^3)/36$ is (choices are: large, close to 0, and close to 36), and the graph of y must have a tangent line that is nearly (choices are: horizontal, vertical). (In both cases, we assume reasonable values for y.)

(b) Verify that all members of the family y = 6 ${(c - x^{2})}^{- \frac{1}{2}}$ $(c - x^2)^(-1/2)$ are solutions of the differential equation y' = $\frac{x y^{3}}{36}$ $(xy^3)/36$ .

y = 6 ${(c - x^{2})}^{- \frac{1}{2}}$ $(c - x^2)^(-1/2)$ => y' = (blank) ${(c - x^{2})}^{- \frac{3}{2}}$ $(c - x^2)^(-3/2)$ .

RHS = $\frac{x y^{3}}{36}$ $(xy^3)/36$ = $\frac{x [6 {(c - x^{2})}^{- \frac{1}{2}})}{36}$ $(x[6(c - x^2)^(-1/2))]/36$ = (blank) ${(c - x^{2})}^{- \frac{3}{2}}$ $(c - x^2)^(-3/2)$ = y' = LHS.

(c) Graph several members of the family of solutions on a common screen.

or

Do the graphs confirm what you predicted in part (a)?

(A) |y'| gets close to 0 if either x gets close to 0 or x gets larger.

(B) As x gets close to 0, |y'| gets larger. As x gets larger, y' gets close to 0.

(C) y' gets close to 0 if either x gets close to 0 or x gets larger.

(D) When x is close to 0, y' is also close to 0. As x gets larger, so does |y'|.

(d) Find a solution of the initial-value problem.

y' = $\frac{x y^{3}}{36}$ $(xy^3)/36$ , y(0) = 12

y = ???

How can I do parts a, b, and c? I think I got the graphing part of c just fine, but I'm not sure. How can I do the rest?

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Answer 1 · 2017-06-23T22:23:55

(a) Close to 0 / Close to 0 / Horizontal / Large / Vertical

(b) Put 6x for both boxes.

(c) www.webassign.net

(d) y = $12 {(1 - 4 x^{2})}^{- \frac{1}{2}}$ $12(1 - 4x^2)^(-1/2)$

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