Differential Equations

The degree of a differential equation is the highest power of the highest-order derivative.
Here, the highest-order derivative is , with power 4 in .
The equation is polynomial in derivatives, so the degree is 4.
Answer: 4.

Step 1: Determine the order of the differential equation. The order is the order of the highest derivative present. The highest derivative is , which is the second derivative. Therefore, the order is 2.
Step 2: Determine the degree of the differential equation. The degree is the highest power of the highest-order derivative, after the equation has been cleared of radicals and fractions with respect to its derivatives. The equation is already a polynomial in its derivatives. The highest-order derivative is , and its power is 2. Therefore, the degree is 2.

Step 1: Identify the form of the differential equation. The equation is in the linear form .
Step 2: Identify the function . By comparing the given equation to the standard form, we find that .
Step 3: Calculate the integrating factor (I.F.) using the formula I.F. = . Step 4: Evaluate the integral and simplify.

Step 1: Differentiate the given equation with respect to . Step 2: This first derivative shows that , which is a constant value since and are constants.
Step 3: Differentiate the result from Step 1 again with respect to . The derivative of a constant ( ) is zero, and the derivative of a constant times a function ( ) is the constant times the derivative of the function. The statement is true.

Separating the variables, the given equation can be written as: Integrating we get, is the required solution.

Since there are two arbitrary constants ( and ), we need to differentiate the relation twice.
Step 1: Differentiate with respect to . Step 2: Differentiate again with respect to . Step 3: Eliminate the constants.
Factor out 9 from the right side of equation (2): Notice that the expression in the parentheses is the original expression for . The required differential equation is .

Concept: Growth proportional to population follows exponential growth: Solution: We use given values to find , then predict future population.
Step 1: Given data Using:
Step 2: Population after another 20 years Total time = 60 years Substitute : Since ,
Step 3: Simplify Approximate: Final Answer: Explanation: Exponential growth means the growth factor remains constant. Using the growth ratio from the first 40 years allows prediction of future population.