How do Muslims care for the elderly?

It is the first God command to Muslims after God worship is to take care of parents and of the elderly in general. You as Muslim is required to:

• take care of them,
• lower your voice when talking to them,
• be patient in dealimg with them,
• follow their advices so far not against Islam teachings,
• help them,
• prefer them in fulfilling their requirements that your wife and children,
• pray for them,
• ask God to bless them and forgive them and dwell them in His Paradise, and
• be always smiling to them and never be bored in listening to them

Quran says (Meaning English translation):

{And your Lord has decreed that you not worship except Him, and to parents, good treatment. Whether one or both of them reach old age [while] with you, say not to them [so much as], "uff," and do not repel them but speak to them a noble Word. (23) And lower to them the wing of humility out of mercy and say, "My Lord, have mercy upon them as they brought me up [when I was] small." (24) Your Lord is most knowing of what is within yourselves. If you should be righteous [in intention] - then indeed He is ever, to the often returning [to Him], Forgiving. (25)}

[Quran, chapter 17, Surat Al-Israa, verses 23-25]

In addition, Islam assures for the elderly the living costs and the required health care and are forgiven from some religious duties and worships if they can't afford it.