How do you find all solutions of the differential equation d^2y/dx^2=x^(−2)?

frobirrimupyx

frobirrimupyx

Answered question

2022-09-13

How do you find all solutions of the differential equation d 2 y d x 2 = x - 2 ?

Answer & Explanation

Marie Horn

Marie Horn

Beginner2022-09-14Added 12 answers

Note that when we integrate things like d x 2 we will be left with dx, so we will need to integrate this twice.
Separating variables:
d 2 y = x - 2 d x 2
Integrating:
d 2 y = x - 2 d x 2
This is analogous to doing d y = x - 2 d x y = - x - 1 + C 1 , but a dx term will still be left.
d y = ( - x - 1 + C 1 ) d x
Integrating again:
d y = ( - 1 x + C 1 ) d x
Add another constant:
y = - ln ( | x | ) + C 1 x + C 2

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