A blue-ribbon government panel opened work yesterday on a desperate mission: find ways to curb the explosive growth of Medicare, the giant program that pays for the health care of 29 million aged and disabled people, and save it from bankruptcy.

The financial troubles of Social Security, Medicare's big brother, are well known. In the past year, there has been endless public debate on the pending cash shortfall in the old-age insurance system, and this week a presidential commission on Social Security is expected to start voting on ways to save the system.

But few people know that stacked up right behind the Social Security problem is a Medicare problem of equal or greater dimensions.

Although it was enacted only 15 years ago, Medicare is the second biggest domestic program in the government, right behind Social Security. Medicare beneficiaries make up more than an eighth of the population. And as with Social Security, its costs are rising faster than its revenues.

Present estimates are that, without congressional action in the interim, the Medicare hospital insurance trust fund will be depleted sometime between 1988 and 1995. The program will be broke.

That day of reckoning was supposed to come sooner, but the tax bill passed earlier this year by Congress put new restrictions on Medicare payments to hospitals, buying an extra three or four years for the hospital fund.

If the economy does extremely well over the next 10 years -- far better than most people expect it will -- the hospital insurance fund is expected to run dry under present tax law and benefit provisions by 1994 to 1995.

If the economy does moderately well (experts call this the "intermediate scenario" and regard it as the most likely possibility), the fund will be bankrupt by 1991.

And if the economy does badly, not even as badly as in the last five years, bankruptcy could come as early as 1988.

Congress will almost certainly prevent this, just as it will almost certainly see to the future of the retirement and disability trust funds out of which Social Security benefits come.

Study groups in the Department of Health and Human Services and in the White House are already looking at possible solutions. Yesterday, the special panel headed by Dr. Otis R. Bowen, former governor of Indiana, appointed by Health and Human Services Secretary Richard S. Schweiker, began work.

Most of the solutions already under study involve further cuts in benefits. Some outside groups and members of Congress have also urged use of income tax revenues to beef up the trust fund, in addition to the revenues from its share of the Social Security tax. And at both ends of Pennsylvania Avenue there has been talk of reorganizing the nation's entire medical reimbursement system to hold down costs.

For many aged and disabled people, the possibility of cuts in medical care is as frightening as potential reduction in monthly Social Security checks. Groups representing them are ready to wage an all-out fight against any severe cuts and they have complained they are not adequately represented on the Bowen panel.

Medicare, enacted as part of Lyndon B. Johnson's Great Society, will lay out over $55 billion this fiscal year, two-thirds for hospital bills and one-third for doctor bills.

In 1981 Medicare paid for one-sixth of all personal health care in the nation. For the aged, it pays about 44 percent of all health bills, including three-quarters of any hospital bills, the main cost it was designed to cover.

The program is divided in two parts -- hospital insurance and doctor insurance. Hospital insurance, the bigger part, is financed by 1.3 percentage points of the 6.7 percent Social Security tax.

The other part of Medicare, "Part B," covering doctor insurance, is financed in part by a $12.20 monthly premium paid by beneficiaries. This premium now covers about a quarter of the doctors' bills covered by the program. The rest, roughly $11.3 billion in calendar 1981, is paid from the Treasury's general fund.

Medicare from its start has been growing faster than most other federal programs, partly because for years hospital costs have been rising much faster than other costs. Medicare itself may have contributed to medical inflation by increasing demands for medical services.

These financial problems, like those of Social Security, are likely to intensify during the next century, when a larger percentage of the population will be over 65.

What to do? These concepts are under discussion:

* Imposing an income limit for benefits, an idea which has already run into so much opposition from the labor movement, other interest groups and on Capitol Hill that it may never be proposed. Medicare has never been considered as a charity program.

* Increasing the basic $260 contribution that a Medicare patient must make to the first 60 days of hospital care.

* Raising the premiums paid by beneficiaries for Part B (doctors' bill) coverage, reducing the level of covered services for both the hospital and doctor insurance programs or raising the out-of-pocket patient contribution toward doctor bills.

* Paying hospitals, and perhaps doctors, by setting a prospective payment in advance and sticking to it.

* Channeling some general revenues from the Treasury into the hospital fund.

* Reorganizing the entire national medical system to do away with the current cost-plus reimbursement system and with some medical tax deductions. Advocates say this would make doctors, hospitals, patients and health insurance companies more cost-conscious and less likely to use services frivolously. This concept is favored by the Reagan administration but others doubt it will work.