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