Question

In Exercises \(15-18,\) solve the differential equation. $$\frac{d u}{d x}=\frac{2}{x^{2}-1}$$

Short answer

The solution to the differential equation is \(u = 2 \ln \left|\frac{x-1}{x+1}\right| + C\).

Step-by-step solution

Identify the Type of Problem

This is a first order differential equation. It is a separable equation given as \(\frac{d u}{d x}=\frac{2}{x^{2}-1}\).

Separate the Variables

Separate the variables by multiplying each side by \(dx\). This gives \( du =\frac{2}{x^{2}-1} dx\).

Integrate

The next step is to integrate both sides. The integral of the left side, \(du\), is simply \(u\). On the other side, using the formula for integrating rational functions, one can say that \(\int \frac{2}{x^2 - 1}dx = 2 \int \frac{1}{x^2 - 1}dx = 2 \ln|x-1| - 2 \ln|x+1|\). Here we have used the formula \(\int \frac{1}{x-a}dx = \ln|x-a|\) . After simplification, this becomes \(2 \ln \left|\frac{x-1}{x+1}\right|\). After integration, we obtain \(u = 2 \ln \left|\frac{x-1}{x+1}\right| + C\) where \(C\) is the constant of integration.

Key concepts

First Order Differential Equation

Solving a first order differential equation often involves finding a function that satisfies a given derivative relationship. In the exercise provided, the differential equation 's solution requires identifying the type of equation and applying the appropriate techniques to isolate the variable and integrate. First order equations are foundational in understanding differential calculus, as they appear in many applications, including physics, engineering, and economics. Solving these equations typically starts with recognizing the form of the function and then proceeding with methods such as separation of variables, integrating factors, or by using a known solution formula.

Separable Differential Equation

A differential equation is considered separable if the variables can be divided in such a way that all instances of one variable are on one side of the equation and all instances of the other variable are on the opposite side. The given exercise demonstrates this property with the relationship . To solve the separable differential equation, you rearrange and multiply each side by the appropriate differential, so as to isolate the du and dx terms, after which you proceed to integrate each side with respect to its own variable, leading to the solution of the differential equation. This property is particularly useful when dealing with equations derived from physics that describe phenomena such as population growth, radioactive decay, or thermal dynamics where variables can naturally dissociate.

Integrating Rational Functions

When faced with the task of integrating rational functions, such as that found in the given exercise , the approach often involves breaking the function into a sum of simpler fractions whose antiderivatives are more straightforward to determine. The integral of a rational function often results in logarithmic expressions, as seen in the decomposition of in the exercise. This method often entails recognizing common integral forms and applying substitution if necessary. Integration of rational functions is a key skill in calculus, as it forms the foundation for resolving more complex integrals found in areas like electrical engineering, aerodynamics, and economic modeling.