Find the particular solution of the differential equation xy(prime)+y=6x^2+8x+2?

xy' + y = 6x² + 8x + 2

Divide by x:

y' + (1/x)y = 6x + 8 + 2/x

This is now of the form:

y' + P(x)y = Q(x)

This is known to have an integrating factor:

U(x) = e^{∫P(x)dx}

P(x) = 1/x

∫dx/x = ln|x|

e^ln|x| = x

Ironically, this is the original equation. The left side integrates to xy and the right side is trivial:

xy = 2x³ + 4x² + 2x + C

Divide by x:

y = 2x² + 4x + 2 + C/x

you have to be really dumb to use the inverse product rule. you cannot solve a problem by going in circles. that's dumb

Easiest approach is by identifying the lefthand side is derivative of xy with respect to x

xy'+y=6x^2+8x+2

LHS=D(xy)=xy'+y

[xy]'=6x^2+8x+2

xy = 2x^3+4x^2+2x+c

y=2x^2+4x+2+c/x

-6=2+4+2+c

c=-14

y=2x^2+4x+2-14/x

using an integrating factor will bring back to this same d.e. so no need to create a viscious cycle

Find the general solution by using the inverse product rule:

xy' + y = 6x² + 8x + 2

x(dy / dx) + y = 6x² + 8x + 2

d(xy) / dx = 6x² + 8x + 2

xy = ∫ (6x² + 8x + 2) dx

xy = 2x³ + 4x² + 2x + C

y = 2x² + 4x + 2 + C / x

Find the particular solution by solving for the constant:

When x = 1, y = -6

8 + C = -6

C = -14

y = 2x² + 4x + 2 - 14 / x

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@ Em

What you just used is the inverse product rule, and you don't know it. You were calling yourself dumb, so you should be more careful with your words.

@ Flamers

There is no reason for me to have five thumbs down. I got the correct answer and it was the only answer that was full and correct at the time.