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

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

If x is close to 0, then y' = #(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 #(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)^(-1/2)# are solutions of the differential equation y' = #(xy^3)/36# .

y = 6#(c - x^2)^(-1/2)# => y' = **(blank)** #(c - x^2)^(-3/2)# .

RHS = #(xy^3)/36# = #(x[6(c - x^2)^(-1/2))]/36# = **(blank)** #(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' = #(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?

(a) What can you say about the graph of a solution of the equation y' =

If x is close to 0, then y' = **(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 **(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

y = 6**(blank)**

RHS = **(blank)**

(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' =

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?

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

(b) Put 6x for both boxes.

(c)

(d) y =

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When ( x ) is close to 0, the behavior of the graph of the solution of the equation ( y' = \frac{xy^3}{36} ) depends on the initial condition of the differential equation.

If the initial condition is such that ( y(0) = 0 ), then the solution will remain at ( y = 0 ) as ( x ) approaches 0.

If the initial condition is such that ( y(0) \neq 0 ), then the graph of the solution will approach a vertical asymptote as ( x ) approaches 0. This is because the denominator of the equation ( y' = \frac{xy^3}{36} ) becomes very small as ( x ) approaches 0, causing the derivative ( y' ) to become large. As a result, the graph of the solution will steeply increase or decrease, depending on the sign of ( y(0) ), near ( x = 0 ).

In summary, when ( x ) is close to 0:

- If ( y(0) = 0 ), the solution remains at ( y = 0 ).
- If ( y(0) \neq 0 ), the solution approaches a vertical asymptote.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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