Differential Equations

The given differential equation is:

To find the order and degree of this differential equation, we need to understand the definitions:

• The order of a differential equation is the highest order of the derivative present in the equation.
• The degree of a differential equation is the power of the highest order derivative after the equation has been made free from radicals and fractions with respect to the derivatives.

Based on this, let's determine the order first:

The highest order derivative in the equation is , so the order of the differential equation is 2.

Next, to find the degree, we must ensure there are no fractional or radical powers involving the dependent variable or its derivatives.

The equation is:

To eliminate the fractional power, we raise both sides of the equation to the power of 4:

Now, the degree of the equation is the power of , which is 4.

Therefore, the order and degree of the given differential equation are:

• Order: 2
• Degree: 4

Thus, the correct option is: 2 and 4.

To solve this problem, we are given two functions and . We need to verify the options provided and determine if all or some of them are satisfied by the given functions.

1. First, let's find the derivative of . Using the product rule, we have:
2. Similarly, calculate the derivative of :
3. We now verify each of the options one-by-one:
4. **Option 1**: Verify .
5. The expression on the right hand side simplifies to:
6. Thus, **Option 1** is satisfied: is true.
7. **Option 2**: Verify .
8. Thus, **Option 2** is satisfied: is true.
9. **Option 3**: Verify .
10. Thus, **Option 3** is satisfied: is true.

Since all three individual verifications hold, the final answer is "All of these".

To solve the problem, we need to determine the probability that three randomly selected numbers from the sets of numbers on cards A, B, and C can form the sides of a right-angled triangle. These numbers range from 1 to 100 on each card.

In a right-angled triangle with sides , , and (where is the hypotenuse), the Pythagorean theorem states:

We will consider each possible combination of numbers selected from the three cards and check whether they satisfy the condition above. However, calculating the exact number of suitable combinations can be simplified using the concept of Pythagorean triples.

We need to find all sets of integers such that:

Enumerating all such combinations manually up to 100 is complex, but we can note common Pythagorean triples within the range. Some basic triples include:

• (3, 4, 5)
• (5, 12, 13)
• (6, 8, 10)
• (7, 24, 25)
• (8, 15, 17)
• (9, 12, 15)
• (9, 40, 41)
• (12, 16, 20)

For the range 1 to 100, there are a total of approximately 16 unique combinations that result in Pythagorean triples due to integer constraint factors and the limitations of the hypotenuse.

Hence, the probability is calculated as:

\)

Total possible outcomes when each of the three numbers can be between 1 and 100:

Favorable outcomes are the valid Pythagorean combinations ( ): approximately 16 combinations.

Therefore, the probability:

This probability does not match any of the provided options exactly, reassuring us that the correct answer is:

None of these