Question

Find the complete integral of p3 +q^3 = 27z.

Answer 1

Let $u= x+ay$ where a is an arbitrary constant. Replacing p by $\frac{dz}{du}$ and q by $a\frac{dz}{du}$

$(\frac{dz}{du})^3+(a\frac{dz}{du})^3= 27 z\\ (\frac{dz}{du})^3+a^3(\frac{dz}{du})^3= 27 z\\ (1+a^3)(\frac{dz}{du})=27z\\ \frac{1+a^3}{z}dz = 27u\\ Integrating \\ (1+a^3)Log \space z= 27 u +b\\ (1+a^3)Log \space z= 27 (x+ay) +b\\$