Solve the differential equation x^3y"'− 3x^2 y" 6xy'−6y = 0.

 3.1     Pull out like factors :

   -3x3y7 + x3y - 6y  =   -y • (3x3y6 - x3 + 6) 

 3.2    Factoring    3x3y6 - x3 + 6 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

 4.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

 4.2      Solve  :    -y = 0 

 Multiply both sides of the equation by (-1) :  y = 0