Question

Integrating factor of
`(x+2y^3)dy/dx=y^2` is:

A. e (1/y)
B. e (-1/y)
C. y
D. -1/y

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Answer 1

the answer is A

(x+2y³)dy/dx = y²
dx/dy = x/y² + 2y

Integrating factor=
IF= e^(∫−1/y^2 dy)
IF= e^(-∫1/y^2 dy)
IF= e^(-(-y^-1))
IF= e^1/y

Answer 2

A) e^(1/y)

Explanation:

Re-write the equation

`\displaystyle (x+2y^3)(dy)/(dx)=y^2\\\\(dy)/(dx)=(y^2)/(x+2y^3)\\ \\(dx)/(dy)=(x+2y^3)/(y^2)\\ \\ (dx)/(dy)=(x)/(y^2)+2y\\ \\(dx)/(dy)-(x)/(y^2)=2y`

Since we have a first-order linear differential equation in the form of
`\displaystyle (dx)/(dy)+p(y)x=q(y)`, then the integrating factor is
`\displaystyle IF=e^{\int {p(y)} \, dy}`. Comparing with the above form, we have
`\displaystyle p(x)=-(1)/(y^2)` and
`q(x)=2y`:

`\displaystyle IF=e^{\int {p(y)} \, dy}\\\\IF=e^{\int {-(1)/(y^2) } \, dy}\\\\IF=e^(1)/(y)`

Thus, A is the correct answer