Question

Finding a General Solution In Exercises \(41-52,\) use integration to find a general solution of the differential equation. $$ \frac{d y}{d x}=\frac{x}{1+x^{2}} $$

Short answer

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

Step-by-step solution

Identify the Differential Equation

The differential equation given is \(\frac{d y}{d x}=\frac{x}{1+x^{2}}\). The key is to recognize that the right-hand side of the equation is in the form that can be directly integrated.

Rearrange the Equation and Integrate

First, rearrange the equation, so we can integrate both sides easily. It will look like this: \(d y=\frac{x}{1+x^{2}} dx\). Now, integrate both sides of the equation, resulting in \(y =\int \frac{x}{1+x^{2}} dx\). This integral can be solved by using the substitution method.

Apply the Substitution Method

Set \(u=1+x^{2}\). Then, \(du = 2x dx \rightarrow du/2 = x dx\).Substitute \(u\) into the equation, we get: \(y=\int\frac{du}{2u}\). Integrate this equation to obtain: \(y=\frac{1}{2}ln|u|+C\).

Substitute Back

Substitute \(u\) back into the equation to obtain the general solution: \(y=\frac{1}{2}ln|1+x^{2}|+C\). This is the general solution to the given differential equation.

Key concepts

Integration

When faced with the task of solving a differential equation, integration is a fundamental tool at your disposal. It is a process that allows you to find a function when its rate of change, or its derivative, is known. In the context of differential equations, integration moves you from the differential form \( dy/dx \) to the antidifferentiated form, which is the general solution, y.

The equation \( dy/dx = x/(1+x^2) \) involves a term that may not be immediately integrable. However, with a clever rearrangement, you create an integral that you can evaluate. To solve for y, you essentially sum up all the tiny changes in y (dy) to get the overall function. The integral \( \int x/(1+x^2) dx \) is the continuous sum of those infinitesimal parts of x over the interval in question, which leads to finding the function y that represents the accumulated change.

Substitution Method

The substitution method in integration is analogous to a 'change of variables'. It simplifies an integral by transforming it into one that is easier to evaluate. By choosing a new variable u that is a function of x, such as \( u = 1 + x^2 \), the differential equation becomes more manageable.

The substitution \( du = 2x dx \) in our problem converts the original integral into a simpler form, \( \int 1/(2u) du \). This new integral represents a basic logarithmic function, which we know how to integrate directly. The substitution method is essential when the integrand (the expression we're integrating) is a composite function, which means it's a function within another function. By substituting, we're able to peel away the outer layer, revealing a simpler problem underneath.

Differential Equations

Differential equations are powerful mathematical tools that relate a function with its derivatives. These equations express physical laws, growth models, and many other phenomena where we deal with rates of change. In the given problem, the differential equation \( dy/dx = x/(1+x^2) \) encapsulates how the rate of change of y with respect to x depends on the value of x.

When you solve a differential equation like this one, you're essentially finding a general solution that includes a family of specific solutions, represented by the constant C. The process usually involves finding the integral of functions or expressions, which, in this case, leads to the logarithmic solution. This general solution tells us about the relationship between x and y without tying us down to one specific scenario or initial condition. In the physical world, this could equate to describing a general law of motion before being given initial speeds or positions.

Calculus

Calculus, the branch of mathematics that deals with continuous change, is the framework in which we understand the problem's dynamics. It encompasses both differentiation and integration. In our differential equation example, we've applied calculus concepts to move from the derivative (differentiation) to the original function (integration).

Calculus allows us to solve problems that involve change, like those seen in physics, engineering, economics, and more. The magic lies in its ability to break down complex, continuously changing phenomena into infinitesimally small pieces that can be analyzed and then summed back up (integrated) to understand the whole picture. In working through this differential equation, we've demonstrated how calculus empowers us to bridge the gap between a rate of change and the function describing that change over time or space.