Source code for Object class can be found here
This source contains implementation of getClass() method (See line 58). hashCode is defined as a function pointer JVM_IHashCode (See line 43).
JVM_IHashCode is defined in jvm.cpp. See code starting from line 504. This in turn calls ObjectSynchronizer::FastHashCode which is defined in synchronizer.cpp. See implementation of FastHashCode at line 576 and get_next_hash at line 530.
Probably, the methods are native for performance and due to practical issues w.r.t implementation.
For e.g., From javadocs, hashCode is typically implemented "by converting the internal address of the object into an integer". This internal address is not available via java sdk and will have to be implemented as a native method.
Please read Is it possible to find the source for a Java native method?. Also read this blog post Object.hashCode implementation. It gives more details. But makes a wrong assertion that hashCode is not generated from object's identity.
Hope it helps.