If you already filed correctly, then no, you can't change anything.
The reason is complicated.
"Correct" filing in your case means you are following the special rules for children whose parents are divorced or separated and share custody. The parent who has physical custody more than half the nights of the year is the only parent who is entitled to claim all the dependent tax benefits -- exemption, child tax credit, earned income credit, dependent care credit, and qualify for head of household. The non-custodial parent can only claim the child as a dependent if
1) there is a divorce order from 2008 or earlier that contains certain specific qualifying language that designates who can claim the child in particular years, or
2) the custodial parent gives the non-custodial parent a signed release form 8332.
In either of those cases, the non-custodial parent can only claim the exemption and child tax credit because EIC, qualify for head of household, and dependent care credit always stay with the custodial parent and can't be transferred.
So, if you are the custodial parent and filed correctly, then you either have a qualifying divorce order, or you gave the other parent a signed release form. You listed the child as a "non-dependent" on the basis of a release form or qualifying divorce order. You can't take back the signed release form, and you can't violate the custody order, so you can't change anything even though your ex did not claim the child.
On the other hand, if you are the non-custodial parent, you can't claim the child without a signed release form even if the other parent did not claim the child. So again you can't change anything.
In the event that you are the custodial parent, and claimed nothing on your tax return, and did not give your ex a signed release (which basically means that, due to ignorance or defiance of the rules you were going to let your ex claim benefits they are not legally entitled to) then you probably can file an amended tax return to claim the other child as a dependent. But please be aware of the rules for next year. If you and your ex ever claim the same child incorrectly, the IRS will investigate both of you.
https://www.irs.gov/uac/about-publication-501
