After five months as prime minister of Canada, Joe Clark enjoys an admiring press and broad public support for a changing national mood that his government brought with it.

Like President Carter, who is scheduled to pay an official visit here Friday and Saturday, the 40-year-old Clark is having trouble with record high interest rates, energy and trade policies and unemployment. Unlike Carter, however, Clark's standing in public opinion polls has not slipped since he assumed office.

With his emphasis on consensus and cooperation, the Progressive Conservative leader's approach stands in sharp contrast to the confrontational politics of his predecessor Pierre Trudeau, whose imperious manner was deeply resented.

Trudeau, meanwhile, seems uncomfortable in his new role of opposition leader, and there are muted calls within the Liberty Party for him to quit and thus open the way for a redefinition of party policies and rejuvenation in the ranks.

While Clark's new government has injected a dose of optimism here, it has yet to tackle the vexing problems that were the source of so much gloom during the final Trudeau years.

The most pressing problem is Quebec separatism, which poses the question of whether Canada can survive as a viable nation if the French-speaking province decides to opt out of the confederation.

A related problem is the nature of the confederation and the function of the central government in Ottawa. Clark was bluntly challenged on this issue last week by a prominent member of his own Progressive Conservative Party, Alberta Premier Peter Lougheed, who threatened to raise oil prices in Canada unilaterally.

The two issues have drained national energies for decades with central government leaders unwilling to tackle them head-on.

When Clark came to power last May after an election that left Canada deeply split along linguistic lines, he was expected to come up with proposals to clarify troublesome federal-provincial relations in a new constitution acceptable to most English-and French-speaking Canadians.

So far, however, the 40-year-old Clark has refused to bite the constitutional bullet. His aides say that it would be premature to talk about a new constitution before a referendum in Quebec next spring on a proposal to make the province sovereign.

Moreover, Clark has decided not to intervene directly in the referendum. He has abandoned Trudeau's plan for staging a counter-referendum in Quebec or the whole country if the outcome of Quebec voting is unclear.

While Quebec residents have welcomed Clark's low-key approach, many Canadians argue that the federal government should advance an alternative constitutional option to the Quebec separatists' sovereignty drive.

The prime minister's argument is that he wants to demonstrate to Quebec residents that the federal system can function in a way that will satisfy their nationalist aspirations.

There was "a pattern of things coming apart in Canada" under the Trudeau government, Clark said in a speech recently. His goal, he said, was "to bring this nation together and the starting step must be to establish a basis for mutual trust."

But when the Quebec government issued its "white paper" on the referendum last Thursday, Clark called it "absolutely unacceptable." A sovereign Quebec in economic association with the rest of Canada "is incompatible with federalism," he said.

Asked what he would do if there is a massive vote in favor of Quebec soverignty, Clark said he had faith in the "good sense" of Quebec voters to reject it.

Diplomats here say that the Conservative government has not developed a clear concept of constitutional changes that would meet the aspirations in Quebec and that would let Quebec Liberal leader Claude Ryan champion the federal cause in the province.

some Liberals belive 1trudeau could play a key role in the Quebec referendum. He has devoted his entire adult life to combating the political independence of Quebec and he still commands considerable following in the province.

But Trudeau's concept of Canada was defeated in the last elections and many Liberals in Ontario and Western Canada believe that the sooner he leaves the better for the party. Several members of the party's national executive are urging an early leadership convention to replace Trudeau.

Liberal insiders here say Trudeau will be replaced next spring, perhaps just after the Quebec referendum, although Trudeau remains publicly committed to staying on.

"My judgment as of now is that I'm the best," Trudeau said last summer.

While Trudeau's assertive style was blamed for federal-provincial quarrels in recent years, Clark is also frustrated in dealing with a powerful and resources-rich Western provinces.

Informed sources say that Clark is expected to back down in the face of Alberta's warning that is planned to increase the price of its oil, now sold at $13.75 per barrel, to the average world level of about $22. The federal government is expected to approve the increase if it is gradually reached over the next two years.

When Alberta, which produces 90 percent of Canada's oil, threatened to raise the price last week, Clark said the increase was "unacceptable." The provinces own mineral rights under Canada's constitution. While the federal government theoretically could sieze Alberta's oil fields under emergency legislation, such an action would be disruptive -- at the least splitting Clark's Conservative Party.

Despite the continuing federal-provincial problems, Clark's reception has been favorable. Since few Canadans expected much from him, most seem satisfied these days.

The only public setback suffered by the government involved a Conservative pledge to move the Canadian embassy in Israel from Tel Aviv to Jerusalem. This provoked Arab anger, accompanied by threats of economic retaliation and Clark dropped the plan.

Although relatively unknown when elected party leader. Clark as prime minister has demonstrated a talent for organization. His Cabinet is small and seemingly efficient He has followed up on his campaign pledges by introducing a freedom of information bill and moving ahead with plans to put the government-run oil company, Petro Canada, in private hands.

His problem in Parliament is that the Conservatives have only 135 of the 280 seats. Unlike an American president, a Canadian prime minister must resign if one of his major bills is defeated in Parliament. Clark thus far has survived three no-confidence votes.