Prime Minister David Cameron's Conservatives may have won the British election and ushered their coalition partner out the door, but that doesn't mean it's all smooth sailing for his government for the next five years.

With influential Euroskeptics clamoring in his own party and a very slim majority in Parliament, Cameron will have a hard time tackling the big headaches looming over his second term: Britain's membership in the 28-nation European Union and the growing movement for Scottish independence.

"He would like to be seen as leading a governing party which is united on a core issue at the center of the political debate, but that's unlikely to happen," said Colin Hay, a British politics professor at the Paris Institute of Political Studies. "It's going to be really tough."

Cameron's Conservatives won an unexpected majority in last week's election, ensuring that he returns to 10 Downing Street with enough power to govern alone. His first term saw Cameron sharing power with the left-of-center Liberal Democrats, who held key positions in a sometimes-awkward coalition government.

Within hours of declaring victory Friday, Cameron re-appointed his four highest-ranking ministers — those heading defense, the Treasury, home and foreign affairs — to their posts. No big surprises are expected when the rest of the new, all-Tory Cabinet is unveiled this week.

The message of stability and continuity is clear: "Keep calm and carry on," as the Times newspaper put it in a headline.

That is easier said than done in Britain's rowdy Parliament, where Cameron does not have full support from his party on the two key topics of Scotland and membership in the EU. The Conservatives now have a tiny majority — holding just over half of the House of Commons' 650 seats — meaning that a dozen defiant Tories could potentially derail important policies.

Rebellion has long simmered in the Conservatives' more right-wing factions, where many want Britain to pull out of the EU. The presence of the pro-EU Liberal Democrats in the government meant that such a move had been out of the question for the past five years.

Radical Conservatives also disagree with their more moderate colleagues over how to deal with the question of Scotland, where the separatist Scottish National Party gained an unprecedented landslide victory in the race for seats in the British Parliament, winning 56 of Scotland's 59 seats.

Cameron and those who back him will do everything to keep Scotland's centuries-old union with England, but some Conservatives are leery of ceding too much power to the north without getting reciprocal benefits for England.

The infighting over those two issues is likely to come to a head in the next two years. Responding to widespread British distrust of Brussels, Cameron has promised to hold a referendum on whether Britain should leave or stay in the EU by the end of 2017. He has maintained that he can negotiate better terms for Britain's EU membership and increase Britain's ability to control the flow of EU migrants to the country.

Cameron plans to meet with restive Conservative legislators Monday to discuss plans for EU reform in an effort to unify the party before discussions with EU leaders about possible modifications to the terms of Britain's membership.

But many rank-and-file Conservatives want much tougher changes, and some have made up their minds that no settlement will be good enough.

"It was a very rebellious parliament the last time," said Simon Usherwood, politics lecturer at the University of Surrey. "You've got some pretty serious Euro-skeptics."

He said it was actually easier for Cameron under the last coalition government.

"Now you can't blame someone else for not being able to get something done," Usherwood said.

Cameron's two major dilemmas are linked: A British withdrawal from the EU would make Scottish nationalists very unhappy and hasten their independence bid.

One area where Cameron will feel less restraint in his new majority government is about welfare cuts, which are expected to hit the poor even harder than those already implemented under his last administration.

The Conservative leader has pledged to reduce the deficit by finding 12 billion pounds ($18.5 billion) to cut from the welfare bill in the next few years. He has not revealed where the money would come from.

"The Lib Dems provided a kind of counterbalance," said Hay. "They helped to hold the Conservative Party together at a time when some on the right were pushing for more cuts to welfare and public services."

Without their coalition partners, the Conservatives face a return of the unpleasant nickname they've long tried to shake: "the Nasty Party."

Some analysts already detect shades of the last majority Conservative government in this one. Like Cameron, former Prime Minister John Major in the 1990s had a fragile majority and struggled to control a party deeply divided over Europe.

But they also believe that Cameron is a political realist. While he himself favors staying in the EU, he's also canny enough not to underestimate his rebellious party members.

"He's pragmatic," Usherwood said. "There's an element of him making it up as he goes along."