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The answer is, maybe.
Though in Kanye West’s song Gold Digger he sings "She got one of your kids, got you for 18 years," implying that a father will have to pay child support until the child turns 18, there are certain circumstances that will require a child support obligor--a mother or a father--to pay child support beyond the child’s 18th birthday.
Payment of child support through high school
As long as a child is enrolled in high school, an obligor must continue to pay child support. Tex. Fam. Code 154.001(a)(1). This portion of the statute is fairly straight forward—in the case of non-disabled children, a child support obligation terminates when the child graduates from high school, even if the child's 18th birthday occurred prior to his graduation date.
Payment of child support beyond the child’s 18th birthday by agreement
To encourage parents to settle their differences amicably, the Texas Family Code allows parents to enter into agreements regarding child support that vary from the child support guidelines, provided that the agreement is in the child’s best interest. Id. at § 154.124. In In the Interest of D.B.J., No. 14-14-002850CV (Tex. App.—Houston [14th Dist.] Feb. 24, 2015), the child turned 18 in February of 2012 and graduated high school the following May. Yet the parents’ divorce decree ordered the obligor parent to pay child support until May of 2016, when the child would be 22. The appellate court determined that such an obligation is not enforceable by contempt but that it is a debt to the obligee parent.
Payment of child support for adult disabled children
Finally, a parent may be ordered to provide support indefinitely to the other parent if their child is disabled. Tex. Fam. Code §§ 154.001(a)(4) and 154.301 et seq. Under the code, a court may order support if the child “requires substantial care and personal supervision because of a mental or physical disability and will not be capable of self-support” and if the disability was known to exist before the child turned 18. Id. at 154.302(a). The amount of support for an adult disabled child is to be determined by examining the existing and future needs of the child, how the child is cared for, the financial resources of the parents, and the financial resources of the adult disabled child. See In re L.J.M., No. 13-13-00367-CV (Tex. App.—Corpus Christi July 3, 2014). See generally In re T.A.N., No. 07-08-00483-CV (Tex. App.—Amarillo Jan. 8, 2010). There, though there were times that the adult child did not require substantial care, the court determined that because of the child’s extensive medical expenses, he was incapable of self-support. A child support order was therefore appropriate.
It should also be noted that the adult child may himself sue his parents for support. See Tex. Fam. Code § 154.303. In In re Sisk, No. 14-13-00785-CV (Tex. App.—Houston [14th Dist.] Oct. 30 2014). There, the appellate court reversed the trial court’s decision dismissing an adult disabled child’s petition that asked the court to order his divorced parents to pay for his support as he was unable to support himself.