Given the first-order linear differential equation:

(
3
𝑥
4
+
1
)
𝑑
𝑦
𝑑
𝑥
=
9
𝑥
5
−
6
𝑥
3
𝑦
.

An integrating factor for the differential equation is:

Hint: An integration factor for
𝑑
𝑦
𝑑
𝑥
+
𝑃
(
𝑥
)
𝑦
=
𝑄
(
𝑥
)
is
𝑒
∫
𝑃
(
𝑥
)
𝑑
𝑥
.

Question

Given the first-order linear differential equation:

(
3
𝑥
4
+
1
)
𝑑
𝑦
𝑑
𝑥
=
9
𝑥
5
−
6
𝑥
3
𝑦
.

An integrating factor for the differential equation is:

Hint: An integration factor for 
𝑑
𝑦
𝑑
𝑥
+
𝑃
(
𝑥
)
𝑦
=
𝑄
(
𝑥
)
  is  
𝑒
∫
𝑃
(
𝑥
)
𝑑
𝑥
.

BlackTom AI · Accepted Answer

We begin by rewriting the given first‑order differential equation into the standard linear form dy/dx + P(x) y = Q(x).
The original equation is (3x^4 + 1) dy/dx = 9x^5 − 6x^3 y. Dividing both sides by (3x^4 + 1) gives dy/dx = [9x^5/(3x^4 + 1)] − [6x^3/(3x^4 + 1)] y.
Bringing the term with y to the left, we obtain dy/dx + [6x^3/(3x^4 + 1)] y = 9x^5/(3x^4 + 1). Thus P(x) = 6x^3/(3x^4 + 1).
An integrating factor μ(x) is defined by μ(x) = e^{∫P(x)dx} = e^{∫ 6x^3/(3x^4 + 1) dx}.
Now compute the integral: let u = 3x^4 + 1, so du = 12x^3 dx and dx terms align as 6x^3 dx = (1/2) du. Therefore ∫ 6x^3/(3x^4 + 1) dx = ∫ (1/2) du/u = (1/2) ln|u| = (1/2) ln(3x^4 + 1).
Exponentiating, μ(x) = e^{(1/2) ln(3x^4 + 1)} = (3x^4 + 1)^{1/2} = sqrt(3x^4 + 1).
Thus, the integrating factor is √(3x^4 + 1), which matches the form shown in the provided option: (3x^4 + 1)^{1/2}.
Why an alternative interpretation would be wrong: any other candidate that omits the square root, or that uses a different power of (3x^4 + 1), would correspond to a different integrating factor not satisfying μ'(x)/μ(x) = P(x). The square root arises specifically because the integral of 6x^3/(3x^4+1) dx yields (1/2) ln(3x^4+1). If one forgot the 1/2 or miscomputed the substitution, they would obtain an incorrect μ(x). Also, selecting a function not dependent on x or failing to recognize the natural logarithm path would be inconsistent with the integrating-factor method for linear equations.
In summary, the calculation shows that the integrating factor μ(x) is indeed √(3x^4 + 1), which corresponds to the provided option.

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