Solving First-Order Differential Equations: A Comprehensive Guide

First-order differential equations are fundamental to many areas of science and engineering. They are equations that involve an unknown function and its first derivative. In this article, we will walk through the process of solving differential equations, specifically focusing on the example: 1x^2 y' x1 - y.

First, let's rewrite the given equation in standard form:

x^2 dy/dx x1 - y

Dividing both sides by x^2 gives:

dy/dx - (1/y) (x1 - y) / x^2

Next, we simplify the right side:

dy/dx (x1 - y) / x^2 (1/y)

Step-by-Step Solution

To solve this first-order differential equation, we will use separation of variables. First, we need to separate the variables y and x on different sides of the equation:

dy/(1-y) (x dx)/(1 x^2)

Now, we integrate both sides:

∫(dy/(1-y) ∫(x dx/(1 x^2))

Integrating the left side:

-∫(d(1-y)/(1-y) ∫(x dx/(1 x^2))

Integrating the right side with the substitution u 1 x^2, we get:

-∫(d(1-y)/(1-y) 1/2 ∫(du/u)

Therefore, we have:

-ln|1-y| 1/2 ln|1 x^2| C

Exponentiating both sides:

1/(1-y) e^(1/2 ln|1 x^2| C)

Further simplifying:

1/(1-y) (1 x^2)^{1/2} e^C

Let A e^C, a constant:

1/(1-y) A (1 x^2)^{1/2}

Solving for y:

1 - y (1 x^2)^{-1/2} A

Thus:

y 1 - (1 x^2)^{-1/2} A

Understanding the Solution

The solution to the differential equation is given by:

y 1 - (1 x^2)^{-1/2} A

This solution involves an arbitrary constant A, which can be determined by initial conditions. The solution shows that the behavior of y depends on the value of x, represented by the term (1 x^2)^{-1/2}.

Conclusion

In this article, we've demonstrated the step-by-step method to solve a first-order differential equation using separation of variables. This method is a powerful tool for solving more complex equations encountered in various scientific and engineering applications. Understanding and mastering these techniques can greatly enhance one's ability to model and analyze real-world phenomena.

Frequently Asked Questions (FAQ)

What is a first-order differential equation?

A first-order differential equation is an equation that involves an unknown function and its first derivative. These equations are widely used in physics, engineering, and mathematics to model dynamic systems.

What is separation of variables?

Separation of variables is a technique used to solve differential equations by rewriting the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the other side. Then, each side is integrated separately.

What is an integrating factor?

An integrating factor is a technique used in solving first-order linear differential equations. It transforms the equation into an exact differential, making it easier to solve. However, this method was not explicitly used in this example.

By understanding these fundamental concepts, you can tackle a wide range of differential equations and apply them to solve real-world problems.