Solving Differential Equations and Finding Summation of Even Numbers

This article will guide you through solving a differential equation using the method of separation of variables and finding the sum of even numbers within a given range. We will start by explaining the process of solving the differential equation and then discuss the summation of even numbers with detailed steps and examples.

Solving the Differential Equation y tan(x) - y

Let's consider the differential equation y tan(x) - y.

Step 1: Rewrite the Equation

We can rewrite the equation in a more workable form as:

dy/dx tan(x) - y

Step 2: Separate the Variables

We can separate the variables by rearranging the equation:

dy/(tan(x) - y) dx

Step 3: Introduce a Substitution

To make the left side easier to integrate, we can use the substitution:

u x - y

This gives us:

du dx - dy > dy dx - du

Substituting for y gives:

dy/tan(u) dx > dy tan(u) dx

Step 4: Integrate Both Sides

Integrating both sides:

int dy/(tan(x) - y) int dx

By substituting u x - y and dy tan(u) dx, we get:

int cot(x - u) dx - int dy int dx

Integrating, we can express this as:

-lnsin(x - u) x C

Step 5: Substitute Back and Simplify

Substituting back u x - y and simplifying:

-lnsin(x - (x - y)) x C

This simplifies to:

-lnsin(y) x C

Exponentiating both sides, we arrive at:

sin(y) e^(-x - C)

Let C e^C, then:

sin(y) Ce^(-x)

Thus, the solution of the differential equation can be expressed implicitly as:

sin(x - y) Ce^(-x)

where C is a constant determined by initial conditions. This represents the general solution to the differential equation y tan(x) - y.

Summation of Even Numbers Between 199 and 1999

To find the sum of even numbers between 199 and 1999, we first identify the even numbers in this range.

The first even number is 200, and the last even number is 1998.

Step 1: Find the Number of Even Terms

The sequence of even numbers is an arithmetic sequence with a common difference of 2. The general form of an even number in this sequence can be written as:

200 2(n - 1)

Setting 200 2(n - 1) 1998, we solve for n:

2n - 2 1798

2n 1800

n 900

Step 2: Use the Sum Formula for an Arithmetic Series

The sum of an arithmetic series can be calculated using the formula:

S_n n/2 [2a (n - 1)d]

where n is the number of terms, a is the first term, and d is the common difference.

Substituting the values, we get:

S 900/2 [2(200) (900 - 1)2]

S 450 [400 1800]

S 450 [2200]

S 990000

Conclusion

Hence, the sum of even numbers between 199 and 1999 is 990000.

Additional Resources

For more detailed explanations and practice problems, consider the following resources:

Table of Integrals Arithmetic Series Tutorial Differential Equations

Understanding these concepts and practicing with various problems will help solidify your knowledge.

Keywords: Differential Equations, Summation of Even Numbers, Separation of Variables