They met Monday in a presidential palace in Bogor - a scenic town about 60 kilometers from Jakarta.
The talks ended with words of cooperation from the American and Indonesian leaders. But the warm welcome Mr. Bush got at the presidential palace was not matched in the streets of Indonesia. In Bogor and elsewhere, there were protests against the presidential visit.
Indonesia is a country with a democratic government, and practices a mostly moderate form of Islam. But there is anger here over U.S. policy in Iraq and Afghanistan, and a perceived American bias in favor of Israel. President Bush has been portrayed by some here as anti-Islam.
Mr. Bush tried to put the protests in the most positive light, saying they were a sign of a vibrant democracy.
"I applaud a society where people are free to come and express their opinion. And it is to Indonesia's credit that it is a society where people are able to protest and say what they think."
He called Indonesia an example of how democracy and modernization can present an alternative to extremism. Mr. Bush said President Yudhoyono understands the power of democracy and knows it is a universal right.
"Your democracy is making Indonesia strong and better able to play a positive role in Southeast Asia and the world."
During a session with reporters, President Bush defended his policies in Iraq and elsewhere in the Middle East. He was asked at one point if more U.S. troops might be needed in Iraq. Mr. Bush said he is getting a lot of advice about Iraq, but cautioned that no decisions are imminent.
"I haven't made any decisions about troop increases or decreases, and won't, until I hear from a variety of sources."
For his part, President Yudhoyono said he favors a multi-facetted approach to ending the conflict in Iraq. He said there must be an emphasis on national reconciliation and bolstering the young Iraqi government. And, he stressed that any long-term solution depends on the ability of other countries to step in and help.
"We have to involve other parties. We have to probably deploy and employ a new set-up of security forces in parallel with, someday at a proper timetable, the disengagement of U.S. military forces and coalition forces from Iraq."
In addition to Iraq, the talks at the Bogor Palace covered a wide range of issues, including North Korea, the search for an end to the Israeli-Palestinian dispute and the development of alternative fuels.
Mr. Bush came to Indonesia Monday from Vietnam, where he attended the Asia-Pacific economic summit in Hanoi, and met with business leaders and medical researchers in Ho Chi Minh City.