Let
𝑦
(
𝑥
)
be a solution to the initial value problem:

𝑑
𝑦
𝑑
𝑥
−
(
9
𝑥
+
8
)
𝑦
2
=
0
,
𝑦
(
−
1
)
=
−
2.

What is the value of
𝑦
(
−
2
)
?

Hints:

Use the method of separation of variables to solve the initial value problem.
∫
𝑥
𝑛
𝑑
𝑥
=
𝑥
𝑛
+
1
𝑛
+
1
+
𝐶
.

Question

Let 
𝑦
(
𝑥
)
 be a solution to the initial value problem:

𝑑
𝑦
𝑑
𝑥
−
(
9
𝑥
+
8
)
𝑦
2
=
0
,
𝑦
(
−
1
)
=
−
2.

What is the value of 
𝑦
(
−
2
)
?

Hints:

Use the method of separation of variables to solve the initial value problem.
∫
𝑥
𝑛
𝑑
𝑥
=
𝑥
𝑛
+
1
𝑛
+
1
+
𝐶
.

BlackTom AI · Accepted Answer

We start by recognizing that the differential equation is separable: dy/dx = (9x + 8) y^2. The variables can be separated as dy/y^2 = (9x + 8) dx. Integrating both sides gives ∫ y^{-2} dy = ∫ (9x + 8) dx, which yields -1/y = (9/2) x^2 + 8x + C. Using the initial condition y(-1) = -2, substitute: -1/(-2) = (9/2)(-1)^2 + 8(-1) + C, so 1/2 = 9/2 - 8 + C = -7/2 + C, hence C = 4. Thus the particular solution satisfies -1/y = (9/2) x^2 + 8x + 4. To find y(-2), evaluate the right-hand side at x = -2: (9/2)(-2)^2 + 8(-2) + 4 = (9/2)(4) - 16 + 4 = 18 - 16 + 4 = 6. Therefore -1/y(-2) = 6, so y(-2) = -1/6.

Note on options: the provided prompt lists a single numeric value as a candidate but does not present other alternatives. If multiple choices were given, each option would be evaluated in turn against this solution path to verify correctness or reveal misunderstandings (e.g., mis-separation, sign errors, incorrect integration constant, or mishandling the initial condition).

SMAT011 Weekly Quiz 8 |LA008

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