Find the general solution of the differential equation

dy/dx = y^4 ( (x+1) / (x-2)(x^2 + 5)) and any other solutions if they exist.

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- Dec 2nd 2009, 12:59 PMgeorgiahelogeneral solution of a differential equation
Find the general solution of the differential equation

dy/dx = y^4 ( (x+1) / (x-2)(x^2 + 5)) and any other solutions if they exist. - Dec 2nd 2009, 07:20 PMmr fantastic
- Dec 5th 2009, 06:20 AMgeorgiahelo
This is where I get stuck, I need a step by step on how to solve the integral of

(x+1)/(x-2)(x^2+5)

It confuses me as to what I do with the top, and the x^2 on the bottom when dealing with partial fractions.can you help? - Dec 5th 2009, 07:30 AMTheEmptySet
Since you have one linear factor and one irreducible quadratic factor with neither factor repeated, the partial fraction decomposition will have the form.

$\displaystyle \frac{A}{x-2}+\frac{Bx+C}{x^2+5}=\frac{x+1}{(x-2)(x^+5)}$

Now multiply both sides by the LCD to get

$\displaystyle A(x^2+5)+(Bx+C)(x-2)=x+1$

expand out the left hand side and collect the powers of x to get

$\displaystyle (A+B)x^2+(-2B+C)x+(5A-2C)=x+1$

Now we want these two polynomials to be equation so we set the coeffients on each power of x equal to get the following system of equations

$\displaystyle A+B=0;-2B+C=1;5A-2C=1$

Solving this system gives

$\displaystyle A=\frac{1}{3},B=-\frac{1}{3},C=\frac{1}{3}$