Question

Find an Euler differential equation of order two with solution:

y=x^3

Answer 1

Eulers sub is t=ln x
am i right?

Answer 2

e^t=x so t=lnx so yep

Answer 3

well then e^(3t) is an answer for eulers rearranged equation for t

Answer 4

x^3=(e^t)^3=e^(3t)

Answer 5

am i clear? i think im not :D

Answer 6

Yep, perfectly clear

Answer 7

well
for example i choose (m-3)(m+3) for equation (t equation)

Answer 8

i mean like:
\[\large y_{t}''-9y_{t}=0\]

Answer 9

Ok

Answer 10

now u just need to convert it to Euler equation (in terms of x)

Answer 11

and u should know an Euler equation
\[x^2 y'' +axy'+by=0\]
with t=ln x becomes to
\[y''+(a-1)y'+by=0\]

Answer 12

Ok, yes, right until now

Answer 13

now u tell me what is the values for a and b

Answer 14

a=1
b=-9

Am I right?

Answer 15

yep
and what about euler equation?

Answer 16

x^2y''+xy'-9y=0 xdd

Answer 17

almost done

Answer 18

u can get many euler equations by choosing (m-3)(m+k)

Answer 19

Ok, thanks @mukushla, great as always

Answer 20

welcome my friend