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Child support is paid by the non-custodial parent
to the custodial parent. Welfare is paid by the state. Where a parent is capable of paying child support and isn't, if the state provides assistance to the custodial parent, the non-custodial parent can be made to repay the state. A person has to apply for and be eligible for welfare, but all custodial parents are entitled to child support (provided they go to court to ask for it). There are a number of different state programs that a person may qualify for - it's not automatic. The term "welfare" includes a lot of different programs, so it would help to know specifically which ones she's receiving, but commonly, that would be Unemployment Compensation, Disability Compensation, or the WIC program (which provides milk).
If you have the child more than half the time, and pay for everything, you can go to court and ask the judge to order your ex to pay you child support. This site has the forms and instructions you need to get started.
This calculator can help you figure out what you might be entitled to, but if her only income is public assistance, that may not be much. Also, even if you have the child more of the time, if you make significantly more than her, the judge could order you to pay her. So, review the calculator carefully before you file anything to verify the numbers.
Just be aware, if you sue her for child support, she's likely to counter-sue for sole custody
. If you don't have a custody order, then you may want to do a preemptive strike and go ahead and file for custody at the same time you file the support order. This site has forms and instructions.