Partial Fraction Decomposition with Arbitrary Constant in $\int\frac{1}{y^4-K^4}dy$.

I need to find the partial decomposition of a fraction which contains an arbitrary constant. This is the final step, or close to final, on a larger non-linear differential equation problem. I need to integrate...

$$\int\frac{1}{y^4-K^4}dy$$

Where $K$ is an arbitrary constant. I'm prettry sure I need to break this fraction up with partial fractions. It's been a long while since I've used partial fractions, and I've tried to brush up, but I can't figure out how to split the fraction up into partials with the unknown constant. I know how I would do it if $K$ where an actual known number, and I've tried just going forward with the way that I know, but I'm getting nowhere. How would I go about splitting...

$$\frac{1}{y^4-K^4}$$

into partial fractions so that I can integrate it?

Solution 1:

Make the ansatz $$\frac{1}{x^4-K^4}=\frac{A}{x-K}+\frac{B}{x+K}+\frac{Cx+D}{x^2+K^2}$$ Multiplying by the denominators we obtain $$1=x^3(A+B+C)+x^2(AK-BK+D)+x(AK^2+BK^2-CK^2)+AK^3-BK^3-DK^2$$ from here you will get an equation system to compute the coefficients