INITIAL VALUE PROBLEMS FOR NON-HOMOGENEOUS ODE

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) Y" + Y = 0.001 X^2 WHERE Y( 0 ) = 0 AND Y'( 0 )= 1.5

THE ANSWER IS Y= 0.002 COS X + 1.5 SIN X + 0.001 X^2 - 0.002

2) Y" + 3Y' + 2.25Y = -10 e^-1.5x WHERE Y( 0 )= 1 AND Y' ( 0 )= 0

THE ANSWER IS Y = ( 1 + 1.5X )e^1.5X - 5X^2

3) Y" + 2Y' + 5Y = e^1.5X + 40 COS 10X - 190 SIN 10X WHERE Y( 0 ) = 0.16 AND Y'(0) = 40.08

THE ANSWER IS Y= 10 e^(-X ) SIN2X + 0.16 e^0.5X + 2SIN 10X

PLEASE EXPLAIN EACH STEP IN NEAT HANDWRITING I AM PREPARING FOR MY FINALS I AM DOING THIS FIRST TIME PLEASE, PLEASE ,PLEASE EXPLAIN IN SUCH A WAY THAT I CAN DO ALL KIND OF THESE QUESTIONS .

I SHALL BE VERY GREATFULL TO YOU DONT FORGET TO LABEL STEPS

PLEASE DO NEAT HAND WRITTING

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