Ruling on slaughtering a cow with the intention of offering the udhiyah and the ‘aqeeqah on behalf of a son and a daughter
Question

I have a son and a daughter, and because of ignorance I did not offer an ‘aqeeqah on their behalf. Now, ten years later, I have realised my mistake, and I am planning to slaughter a cow on the upcoming Eid al-Adha, to be both an udhiyah and an ‘aqeeqah for both children, seeing that a cow is acceptable on behalf of seven people. I will divide it into one seventh for the daughter, two sevenths for the son, and four sevenths for the udhiyah. But I do not know what is the ruling on that. I have become more confused because of some video clips and articles by some scholars, some of whom say it is permissible and some say it is not allowed. I hope that you can clarify.
Praise be to Allah.Slaughtering a cow with the intention of some of it being for the ‘aqeeqah and some for the udhiyah is a matter concerning which the scholars differed. The Hanafis and Shaafa‘is regard it as permissible. 
Ibn ‘Aabideen al-Hanafi said, concerning the permissibility of making a joint intention in such cases: 
It is permissible regardless of whether the sacrifice is obligatory on all or some of them, and whether the reason for it is one reason or not, such as if the reason is as a sacrifice (udhiyah) or as a result of being prevented from proceeding to Makkah (after having entered ihram), or as a penalty for hunting game or for shaving (when in ihram), or because one is doing tamattu‘ (‘umrah followed by Hajj, exiting ihram in between) or qiraan (‘umrah followed by Hajj, without exiting ihram in between). This is different from the view of Zufr, because the aim in all cases is to draw closer to Allah. The same applies if the reason of some of them is to offer ‘aqeeqah for a child who was born before that, because the ‘aqeeqah is an act by which one intends to draw closer to Allah by showing gratitude for having a child. 
In al-Fataawa al-Fiqhiyyah al-Kubra by Ibn Hajar al-Haytami ash-Shaafa‘i (4/256) it says: 
If he slaughters a camel or a cow for seven reasons, including udhiyah and ‘aqeeqah, and the rest are expiations for things such as shaving whilst in ihram, that is acceptable, and it does not come under the heading of superimposition, because each of the seven parts of the camel or cow will be for a particular reason, and that is valid. End quote. 
The Hanbalis disallowed making additional intentions when offering the ‘aqeeqah in all cases, so in their view it is not acceptable to slaughter a cow or camel unless it is all for one ‘aqeeqah, for one child. It says in Sharh Muntaha al-Iraadaat (1/614):
It is not acceptable to slaughter a camel or a cow for an ‘aqeeqah, unless the entire animal (is intended for that ‘aqeeqah). End quote. 
In al-Mubdi‘ fi Sharh al-Muqni‘ (3/277) it says:
Our view is that it is not acceptable (in the case of ‘aqeeqah) for it to be done as a share of an animal, and it is not acceptable unless the entire camel or cow is intended for that reason. End quote. 
What is more likely to be correct is that it is not permissible to make a shared intention with regard to the ‘aqeeqah, because there is no report to suggest that a shared intention is permissible in that case, unlike the udhiyah. Moreover, the ‘aqeeqah is done as a ransom for the child, so it must be one for one. So it is not acceptable to offer anything except a complete cow or a complete camel or a complete sheep. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in ash-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘ (7/428):
“A camel or a cow may be offered on behalf of seven (people)” – an exception to that is made in the case of the ‘aqeeqah, for which a camel can only be offered on behalf of one individual. However, offering a sheep is preferable, because the ‘aqeeqah is a ransom for one soul, and the ransom must be one for one, so it the ransom of a soul for a soul. If we say that the camel is acceptable for seven, then it would be a ransom of one soul for seven souls. Hence they said: In the case of ‘aqeeqah, the entire animal should be offered, otherwise it is not acceptable. 
If a man has seven daughters, all of whom need the ‘aqeeqah, and he slaughters a camel on behalf of all seven, that is not acceptable. 
But is it acceptable on behalf of one? Or do we say that it is not prescribed to do this act of worship in this manner, and we should bring several animals and slaughter an ‘aqeeqah on behalf of each daughter? The latter is more correct. We should say that it is not valid to offer one sacrifice (on behalf of seven daughters), because that is contrary to what is narrated in sharee‘ah. So he should slaughter a sheep on behalf of each one, and this camel that he slaughtered belongs to him, and he may sell its meat, because it is clear that it is not valid as an ‘aqeeqah. End quote. 
See also the answer to question no. 82607 
Based on that, it is not valid for you to slaughter a cow to be both an udhiyah and ‘aqeeqah on behalf of your two children, and you have to offer a sheep as ‘aqeeqah, because that is better. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in ash-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘: … with regard to the ‘aqeeqah, a sheep is better than an entire camel, because it is what is mentioned in the Sunnah, so it is better than a camel.
End quote. 
So you should slaughter two sheep on behalf of your son and one sheep on behalf of your daughter. 
With regard to the udhiyah, you have the choice between camels, cows, and sheep. The best is a camel, then a cow, if you offer the entire animal without sharing, then a sheep. This has been explained in detail in fatwa no. 45767 
And Allah knows best.