Am I accountable for the sins of what my brother receives or sends—music, songs, and time-wasting—via the WhatsApp application that I downloaded for him, and what is the solution?

If you do not know that your brother will use the program for something forbidden, then there is no blame upon you for downloading it, and you do not sin if your brother uses it for something forbidden. However, if you know from the outset that he will do so, then you do sin by downloading it for him, because you have assisted him in disobedience. And whoever causes a sin shares in its burden. The Messenger of Allah (peace and blessings be upon him) said: "And whoever introduces a bad practice in Islam will bear its sin and the sin of those who act upon it after him, without anything being diminished from their sins." The solution is for you to advise your brother. If he insists on using it for something forbidden, and you are able to remove the program from his device without harm, then you must do so. Otherwise, you have done what you could.