Compute the general solution of the linear inhomogeneous differential equation y' = $\displaystyle \frac{-y}{1 + x}+ 3x$ .

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- Feb 16th 2009, 12:49 PMronaldo_07[SOLVED] differential equations
Compute the general solution of the linear inhomogeneous differential equation y' = $\displaystyle \frac{-y}{1 + x}+ 3x$ .

- Feb 16th 2009, 01:27 PMronaldo_07
I fond the ingegrating factor to be 1+x and now I dont know where do to from there

- Feb 16th 2009, 01:36 PMronaldo_07
I solved it to be y= $\displaystyle \frac{\frac{3x^2}{2}+x^3 +C}{1+x}$

- Feb 16th 2009, 01:46 PMHallsofIvy
It doesn't help to find the integrating factor if you don't know what an "integrating factor"

**is**! Don't memorize formulas, learn concepts!

An integrating factor for a linear differential equation of the form dy/dx+P(x)y= f(x,y) is a function of x, $\displaystyle \mu(x)$ such that multiplying both sides of the equation by it converts the left side into a "perfect" derivative so that the equation becomes: $\displaystyle \mu(x)dy/dx+ \mu(x)P(x)y= \frac{d(\mu(x)y)}{dx}= \mu(x)f(x,y)$.

If (1+ x) really is an integrating equation (I haven't checked that) then multiplying 1+x on both sides converts the equation $\displaystyle \frac{dy}{dx}+ \frac{1}{1+ x} y= 3x$ to $\displaystyle \frac{d((1+x)y}{dx}= 3x((x+1)$. Now integrate both sides with respect to x.