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FIND THE GENERAL SOLUTIONS OF EULAR CAUCHY EQUATIONS

BEFORE DOING THESE QUESTIONS PLEASE IT IS MY HUMBLE REQUEST TO SOLVE ALL QUESTIONS STEP BY STEP PLEASE LABEL EACH STEP I AM DOING THIS FIRST TIME AND I DONT KNOW HOW TO DO IT I AM POSTING MY QUESTION 4TH TIME FIRST THREE TIMES I DIDNOT GET MY FULL ANSWERS AND DIDNOT GET MY ANSWERS STEP BY STEP PLEASE AFTER ONE STEP GIVE SOME EXPLANATION AND DO IN THAT WAY THAT I CAN DO ALL THESE KIND OF PROBLEMS AGAIN PLEASE DO NEAT HANDWRITING AND EXPLAIN EACH STEP AND LABEL THEM AS 1, 2 3 ETC.

1)    X^2 Y" + 1.5 XY' -0.5Y =0 ( FIRST TELL THAT HOW TO IDENTIFY THAT THIS IS EULAR CAUCHY EQUATION )

IT HAS AUXILIARY EQUATION

M^2 + 0.5 M - 0.5 = 0 ( HOW WE OBTAIN THIS )

THE ROOTS ARE 0.5 AND -1 , HENCE THE BASIS OF SOLUTIONS FOR ALL POSITIVE X IS Y1= X^0.5 AND Y2 = 1/X AND GIVES THE GENERAL SOLUTION

Y = C1 . SQRT(X) + C2/ X    ( HOW THIS WILL BE OBTAIN)

2) X^2. Y" + 0.6 XY' + 16.04 Y =0    ( HOW WE IDENTIFY)

THE AUXILARY EQUATIONS IS M^2 - 0.4M + 16.04Y= 0 (HOW WE OBTAIN )

THE ANSWER IS Y = X^0.2 ( A COS (4 ln X ) + B SIN (4ln X)

3)    X^2.Y" - 5XY' + 9Y =0

THE AUXILARY EQUATION IS M^2 -6M +9 =0

THE SOLUTION IS ( C1 + C2lnX ) X^3

PLEASE EXPLAIN THESE QUESTIONS IN SUCH A WAY THAT I CAN DO ALL TYPE OF QUESTIONS