Question

Solving a Differential Equation In Exercises \(1-10\) , solve the differential equation. $$ \left(1+x^{2}\right) y^{\prime}-2 x y=0 $$

Short answer

The solution to the given differential equation is \(y = \ln |1 + x^{2}| + C\).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is a first order homogenous differential equation of the form \(M(x,y)dx + N(x,y)dy = 0\), where \(M = (1+x^{2})y\) and \(N = -2xy\).

Convert into Variable Separable Form

To solve this type of equation, it can be rewritten in the form that separates the variables x and y. Let’s divide the equation by \((1+x^{2})y\) to yield \(dy/dx - (2x)/(1+x^{2}) = 0\). This equation can be now rewritten as \(dy/dx = (2x)/(1+x^{2})\).

Integrate Both Sides

To find the solution, we find the integral of both sides respect to x. For the left side, the integral is simply y. For the right side, using the power rule for integration, we have \(\int (2x)/(1+x^{2}) dx = \ln |1 + x^{2}| + C\), where C is the constant of integration.

Write Final Equation

Now we can rewrite the equation to express y in terms of x. So the solution of the given equation is \(y = \ln |1 + x^{2}| + C\).

Key concepts

First Order Homogeneous Differential Equation

In the study of differential equations, a first order homogeneous differential equation is an important class that is characterized by its simplicity and symmetry. This type of equation is defined by the presence of derivatives of only one variable and relies on the principle that all terms can be transformed into a function that equals zero.

Generally, such equations take the form \(M(x,y)dx + N(x,y)dy = 0\) and are considered homogeneous if the functions M and N are of the same degree when considering both variables x and y. When an equation is labeled as homogeneous, it implies a certain uniformity in the behavior of solutions, offering elegant methods for finding these solutions.

To effectively approach these problems, recognizing the homogenous nature of a differential equation becomes crucial, as it determines the most suitable solution strategy — one that often involves transforming the equation into a format that allows for the separation of variables.

Variable Separable Form

The variable separable form is a technique commonly used to solve first order ordinary differential equations. This approach is based on the idea of separating and grouping all terms involving one variable on one side of the equation and all terms involving the other variable on the opposite side. In essence, it breaks down a complex relationship into simpler chunks that are easier to work with.

When an equation is rewritten into this separable form, it enables the use of straightforward integration on both sides of the equation — a method that significantly simplifies the process of finding a solution. For example, transforming an equation like \(dy/dx - (2x)/(1+x^{2}) = 0\) into \(dy/dx = (2x)/(1+x^{2})\) aligns it with the criteria for using the variable separable method.

Understanding how to manipulate an equation into this form can be the key step in making an intimidating differential equation more manageable. It allows for a strategic application of integration to each variable independently, ultimately leading to the identification of the function y in terms of the variable x.

Integration of Differential Equation

The process of integration of a differential equation refers to finding the antiderivative or integral of functions within the equation. This step is essential in the journey of uncovering the underlying function that the differential equation represents. Essentially, integration moves us from the derivative of a function (which provides the rate of change) back to the original function itself.

In the case of variable separable forms, integration can be applied to both sides of the equation. As seen in the example \(dy/dx = (2x)/(1+x^{2})\), integrating the right-hand side with respect to x could involve techniques such as the power rule for integration, which allows for the determination of antiderivatives. The act of integration results in an antiderivative plus a constant of integration, typically represented by C. Thus, if we find \(\int (2x)/(1+x^{2}) dx = \ln |1 + x^{2}| + C\), we have succeeded in integrating the differential equation.

The aim is to reach a point where the function is expressed in a closed form, meaning y is explicitly described in terms of x, along with any integration constants. This gives us a clear understanding of the solution and how it behaves for varying values of x, allowing deeper exploration into the specific dynamics described by the differential equation.