The prime minister said on BBC1's Andrew Marr Show that he would "strain every sinew" and "redouble all our efforts" to boost growth, after figures this week revealed the UK had plunged back into recession.
"What I have to do is convince people that we are taking the right long-term decisions to deal with problems in our country that have existed for many years," Cameron said.
Pressed on the fact the US economy was growing under a far less stringent austerity regime, Cameron said Washington's plans were for "tougher austerity measures" and it did not have the eurozone crisis on its doorstep.
Pointing to the recent economic woes in Spain and the Netherlands, Cameron indicated that eurozone countries would have to decide whether they wanted to continue with a single currency and economic policy.
"I don't think we are anywhere near halfway through it because what's happening in the eurozone is that there is a massive tension between a single currency that countries are finding very difficult to adapt to.
"I think it's going to be a very long and painful process in the eurozone as they work out do they want a single currency with a single economic policy and all the things that go with it or are they going to have something quite different. That's what they have to decide."
He compared the situation on the continent with that in the UK. "The way it works in the United Kingdom is that different parts of the UK support each other and there isn't that in Europe and that's their problem, or one of their problems."
The prime minister defended his government's economic plan amid the Conservatives' worst polling since entering government, days before local elections take place. A YouGov poll for the Sunday Times showed support for the Tory party had fallen to 29%, with Cameron's personal approval rating also suffering a blow. Labour was on 40%, the Liberal Democrats 11% and Ukip 10%.
A separate poll for the Sunday Express found eight out of 10 voters believed the coalition had lost touch with ordinary families.
The shadow chancellor, Ed Balls, warned that Britain faced a "lost decade" of sluggish growth and high unemployment if the government failed to change course on the economy. He told the Observer that lacklustre growth could be an issue for a very long time. "I am not predicting years of recession, although we could well be in for a long lost decade-style period of slow growth and high unemployment," he said.
Cameron said on Marr that the UK was going through "a very difficult, painstaking process" of rebalancing the economy because the old model of growth that focused on financial services in England's south-east was broken.
He said the private sector was growing, "but not as fast as I would like", and vowed to step up efforts "to make sure that whether it's bank lending, whether its manufacturing industry, whether it's regional policy … work programmes to help millions of people with training, make sure all these things are delivering".
Balls said on Sky News: "David Cameron is saying the economy is the most important thing but he's doing nothing about it. He's nothing to say, his plan's not working and he's ploughing on regardless."
Scotland's most senior Roman Catholic, Cardinal Keith O'Brien, accused Cameron of acting immorally by favouring the rich, and denounced the prime minister's opposition to a "Robin Hood tax" on financial institutions.
The Sunday Times rich list for 2012 reveals that Britain's wealthiest people are richer than they have ever been at a time when the rest of the country is steeped in the worst recession since the 1930s, raising fears about the growing gap between the most affluent and the rest of society. The 1,000 most well-off men and women in the country increased their wealth by 4.7%, accruing a combined fortune of £414bn.
A Downing Street spokesman said the prime minister was "determined to help people" struggling with their finances.
