Is the word ajude the same in Brazilian and European Portuguese?

Yes

The only difference between "ajude" in Brazilian Portuguese and Continental Portuguese is pronunciation. In Brazil, the unstressed final "e" is typically pronounced as a high front vowel [i], making the word sound like a-ju-dji. In Portugal, the unstressed final "e" is pronounced as a near-close central unrounded vowel [ɨ], which sounds like a muffled or "swallowed" sound, almost like a very short, neutralized "e."

Brazilian Portuguese Examples

1. Espero que ele me ajude com a mudança. (I hope he helps me with the move.)
2. Por favor, ajude o seu irmão. (Please, help your brother.)
3. Não quero que ninguém ajude agora. (I don't want anyone to help right now.)
4. É necessário que você ajude no projeto. (It is necessary that you help with the project.)
5. Peça para que ela ajude na limpeza. (Ask her to help with the cleaning.)

Continental Portuguese Examples

1. Espero que ele me ajude com a mudança. (I hope he helps me with the move.)
2. Por favor, ajuda o teu irmão. (Please, help your brother. — Note: In Portugal, the informal "tu" form "ajuda" is much more common than the "você" form "ajude" when speaking to peers/family.)
3. Não quero que ninguém ajude agora. (I don't want anyone to help right now.)
4. É necessário que tu ajudes no projeto. (It is necessary that you help with the project. — Note: Using the "tu" conjugation "ajudes" to express the same idea.)
5. Pede para que ela ajude na limpeza. (Ask her to help with the cleaning. — Note: Using the "tu" imperative "pede" instead of the "você" form "peça".)