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Find the general solution?

Find the general solution of (x+1)^2*y"+3(x+1)y'+0.75y=0 that is valid in any interval not including the singular point.

Answer: y=c1*abs(x+1)^(-1/2)+c2*abs(x+1)^(-3/2)

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Karina R. | Explains it AllExplains it All
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This is just one of those polynomial cases. You start by assuming that y is of the form (x+1)^r, calculate y' and y'', plug them back in, factor what you get, and solve the r equation that you get so that the whole thing can be zero. You get more than one value of r, so the general solution is a linear combination of both values.