schaukeln
to swing
To move back and forth on a swing.
Die Kinder schaukeln begeistert auf dem Spielplatz.
The children are happily swinging on the playground.

to swing / rock
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to swing
To move back and forth on a swing.
Die Kinder schaukeln begeistert auf dem Spielplatz.
The children are happily swinging on the playground.

The verb 'schaukeln' (to swing/rock) is directly connected to the noun 'die Schaukel' (the swing). A 'Schaukel' is the piece of equipment you swing on. If you know one of the words, you can easily work out the other.
Die neue Schaukel im Garten ist toll, die Kinder wollen nur noch schaukeln.
The new swing in the garden is great; the children only want to swing now.
You can tell from the subject whether it refers to people on playground equipment or to a gentle movement. If people, especially children, are the subject, it is usually about a swing. If a boat or ship is the subject, it refers to a rocking motion.
Das Kind schaukelt so hoch es kann.
The child is swinging as high as they can.
In the meaning of 'wiegen' (to rock) or 'schwanken' (to sway), 'schaukeln' (to rock/sway) is often combined with things that are moved by external forces such as water or wind. Typical phrases are 'auf den Wellen schaukeln' (to rock on the waves) or 'im Wind schaukeln' (to sway in the wind).
Die Äste des Baumes schaukelten sanft im Wind.
The branches of the tree swayed gently in the wind.
The colloquial expression 'das Kind schon schaukeln' has nothing to do with children or swings. It means 'das schaffen wir schon' (we'll manage it) or 'das wird schon klappen' (it'll work out) and expresses confidence.
Die Prüfung wird schwer, aber keine Sorge, wir werden das Kind schon schaukeln.
The exam will be difficult, but don't worry, we'll manage it.
to manage / create
'schaffen' and 'schaukeln' are similar only in a special colloquial expression: 'das Kind schon schaukeln' (to manage it somehow). In that case, 'schaukeln' means that you will somehow manage a difficult matter. 'schaffen' is the normal word for a task or challenge. On its own, 'schaukeln' otherwise usually means a back-and-forth movement, for example on a swing. The confusion arises because the idiom works like 'schaffen'. Without this idiom, it is better to use 'schaffen' for tasks.

Keine Sorge, wir schaukeln das Kind schon und finden bis morgen eine Lösung.
Don't worry, we'll manage it somehow and find a solution by tomorrow.
Aylin schafft die Prüfung, wenn sie diese Woche noch regelmäßig lernt.
Aylin will manage the exam if she keeps studying regularly this week.