1Using CauchyEuler Method find the general solution for X3 y

1.Using Cauchy-Euler Method, find the general solution for:

                        X³ y

Solution

2) Auialary equation is

m = 2, 5

Hence general solution is y = Ae^2x+Be^5x+x^4/6-2x^2+e^x

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3) y\"+y\'/x = 1

Put y\' = v

y\" = v\'

Hence v\'+v/x =1

This solution is ve^x = integral of e^x

v(e^x) = e^x+C

Now substitute y\' = V

y\' (e) = e+C

Use the equation y\'(1) = -1/2

-1/2(e) =C+e

C = -3e/2

y\'e^x =-3e/2+e^x

integrate to get

y\' = -3e/2 (e^-x) + 1

y = 3e^x+1/2+x+C\'

Use y(1) =1

1 = 3e+C\"+1

C\' =-3e

Hence solution is

y = 3e^x+1/2+x-3e