One month after its creation, Phodesia's new multiracial government is beset by major military, political and economic problems as it struggles to find a broad base of support at home and abroad.

Notwithstanding signs of encouragement from South Africa and of growing sympathy in some Western circles, the transitional government still stands very much alone.

There is "tremendous apprehension" among whites, in the words of Prime Minister Ian Smith, about the changes that must come and mounting pressure from blacks to see quick evidence that a new era is dawning.

"To win over the black people, we have to be seen to be making changes. We can't just sing the praises of the (internal) settlement, we have to do something," said the new black cominister of justice, Byron Hove.

It is precisely Hove's outspoken demand for such immediate signs of change as more and higher-ranking black policemen and judges that has upset an already jittery white population.

Smith and his white ministers, whatever their private feeling about the coming of black majority rule, are touring the country pleading with the 260,000 remaining whites to wait the final outcome before jumping the gun" and leaving.

The black leaders of the transitional government are just as uncertain about the new situation. Still surrounded by white civil servants, secretaries and guards and sitting apart in bare offices, they are grouping for ways to illustrate their new authority to the 6.7 million blacks.

Clearly the dilemma of this young, fragile experiment in multiracial cooperation is how to satisfy the country's two fundamentally different populations filled with contrasting fears and hopes. The new ruling Executive Council must also convince skeptical Western and African states that it has a true claim to representing the aspirations of the majority of Rhodesia's people - whites and blacks, including those who belong to the guerrilla alliance, the Patriotic Front.

Moreover, it must do all this while waging a war that by most accounts is going badly and getting worse. "It's bound to," said P. K. Van Der Byl, the dapper, hardliner who is the white cominister of foreign affairs. "Those who don't think they can win an election are making their last big effort," he said.

The war is costing nearly $2 million a day and even the reported reintroduction of South African military forces in the southern border area is unlikely to provide much relief.

The military situation is exceedingly complicated. Military analysts in constant contact with the troops speak of sharply declining morale among white officers. "Nobody," said one analyst, "wants to be the last white man to die for Zimbabwe." Zimbabwe is the African name for Rhodesia.

In some areas, there is apparently a virtual halt in fighting, in part at least because local commanders have been told not to seek contact with the guerrillas. In part, too, it seems there are an increasing number of "local arrangements" between the army and guerrillas. The rebels are split into factions which are often as opposed to each other as to the Salisbury government.

While the white-led army is in full flux, a growing number of white farmers are said to be abandoning their homesteads and taking the "chicken run" to South Africa.

Even Allan Savory, a staunch Rhodesian and leader of the tiny white Liberal Party here, has had to close down one of his ranches and buy a house in Salisbury for a white manager of another to give the man and his family a respite from the daily dangers of the countryside.

In fact, Rhodesians say that no prudent white will go outside the main towns unarmed these days for fear of ambush.

It is universally agreed that a cease-fire is essential if this experiment in joint white-black government is to survive until the proposed elections for a black majority government sometime next fall.

The black leaders of the Executive Council, Bishop Abel Muzorewa, Rev. Ndabaningi Sithole and Chief Jeremiah Chirau, express confidence they can pull it off and tell of daily contacts with guerrillas now inside the country to arrange an end to the fighting. Outside observers, mindful of the military salippage, remain doubtful and await the outcome of the call for cease-fire to be convinced.

Just how the Executive Council can hold elections amid the present uncertainties is puzzling. If elections for the formal establishment of a black Zimbabwe government are to be accepted abroad, some means must be found to get the guerrillas out of the bush and to the ballot box. Yet their leaders, who have denounced the internal settlement, have scorned overtures to return.

An Anglo-American proposal intended to bring together all black nationalist leaders for an election appears to be going nowhere. private efforts aimed at getting one or both of the Patriotic Front leaders, Joshua Nkomo and Robert Mugabe, back show no sign of success.

Even if no progress is made with the guerrillas, Smith says he and his black colleagues are determined to go ahead with elections and says "any impartial, just observation" by outside observers is welcome.

Smith, who boasts of the economic accomplishments during 12 years of international isolation, also contends that economic problems are the most serious threat to the country, particularly for the rising number of unemployed blacks.

Despite - or because of - economic sanctions, Rhodesia has one of the broadest-based economies in Africa. But the steady exodus of whites and their capital together with the war's drain on manpower and other uncertainties underlies the country's present-day economic crisis.

In its statement Tuesday turning down the Anglo-American proposal for an all-party conference, the Executive Council said:

"We are satisfied that collectively we have obtained the support of the overwhelming majority of the population for the Salisbury [internal] agreement and we ask that the British and United States governments should consider putting this to an early test.

"We are not concerned so much with formal recognition as we are with countering the effects of the world economic recession by an easing of the restrictions on our trade so that we can alleviate unemployment among blacks and proceed to majority rule with a sound and strong economy."

Despite the multitude of problems weighing down upon the new government, Rhodesia under white and black hands seems grimly determined to press on against all obstacles, just as it did during the 12 years of white minority rule, in defianceof the outside world.