Question

Find the general solution of the following differential equations. $$u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}$$

Short answer

Question: Find the general solution of the first-order differential equation: $$u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}$$ Answer: The general solution of the given differential equation is: $$u(x)=\frac{1}{2}\ln{\frac{|x^2+4|}{x^2+4}}+C$$

Step-by-step solution

Identify the type of differential equation

We have a first-order differential equation of the form: $$u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}$$ This is a first-order, separable differential equation.

Separate the variables

To separate the variables, we need to rewrite the equation in the form of \(u^{\prime}(x) = f(x)\). In this case, we have: $$u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}$$

Integrate both sides

Now we'll integrate both sides of the equation with respect to x: $$\int u^{\prime}(x) \, dx = \int \frac{2(x-1)}{x^{2}+4} \, dx$$ Let's focus on the integral on the right side.

Apply substitution to the integral

We'll perform a substitution to simplify the integral. Let \(t=x^2 + 4\). Then, \(dt = 2x \, dx\). So, the integral becomes: $$\int \frac{2(x-1)}{t}\frac{dt}{2x}= \int \frac{(x-1)}{t} \, dt$$

Complete the integration

Now, we can perform the integration: $$u(x) = \int \frac{(x-1)}{x^2+4}\, dx = \int \frac{(x-1)}{t} \, dt$$ Expand the fraction in the integral: $$u(x) = \int \frac{x}{t} \, dt - \int \frac{1}{t} \, dt$$ Here, perform the integration to get: $$u(x)=\frac{1}{2}\ln|x^2+4|-\frac{1}{2}\ln{t}+C$$

Substitute back and simplify

Swap back the substitution we made earlier, \(t = x^2 + 4\): $$u(x)=\frac{1}{2}\ln|x^2+4|-\frac{1}{2}\ln{x^2+4}+C$$ Combine the constants: $$u(x)=\frac{1}{2}\ln{\frac{|x^2+4|}{x^2+4}}+C$$ The general solution of the given differential equation is: $$u(x)=\frac{1}{2}\ln{\frac{|x^2+4|}{x^2+4}}+C$$

Key concepts

General Solution

A general solution to a differential equation is a formula that encompasses all possible solutions. It includes an arbitrary constant, often represented by "C", which allows for a continuity of solutions, each corresponding to different initial conditions. In this context, finding the general solution of the differential equation \[u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}\] involves crafting a function, denoted as \(u(x)\), that represents the behavior of the differential equation across a range of values.
The general solution provides a wide scope of possible outcomes based on varying 'C' values, enabling it to align with specific initial conditions you might encounter.

• The constant \(C\) does not change the derivative, keeping the differential equation satisfied.
• Solving for the general solution often requires integration, capturing every potential behavior of the differential we began with.

Variable Separation

Variable separation is a method used to solve first-order separable differential equations, where you can separate the variables on either side of the equation. This technique essentially allows you to rearrange the equation such that each variable and its differential coefficient is on separate sides.
In our example: \[u^{\prime}(x)=\frac{2(x-1)}{x^{2}+4}\]we begin by securing all terms involving \(x\) on one side, freeing \(u'(x)\) and relating it distinctly to its own variable.
This results in a setup that gives us an equation prime for integration.

• Separation leads to \(u^{\prime}(x) \, dx = \frac{2(x-1)}{x^2+4}\, dx\), preparing for the next integration step.
• Effectively separates an equation into distinct parts, making it simpler to solve.

Substitution Method

In tackling integrals, the substitution method is a handy tool that can simplify complex expressions. It involves replacing complicated parts of an integral with a substitution variable, which makes the integration process more manageable.
For the differential equation at hand, when integrating \[\int \frac{2(x-1)}{x^2+4} \, dx\], we can apply a substitution \(t = x^2 + 4\) which simplifies calculations.
This step changes the variables and simplifies the expression to something more straightforward, altering the integration path constructively.

• Substitution helps in transforming a difficult integral into an easier one.
• Makes it feasible to handle by introducing a new "t" variable to lighten computations.

Integration

Integration is the inverse operation of differentiation. It's an essential tool in finding the general solution of differential equations; in this context, it determines the behavior over an interval by summing infinitesimal changes.
By integrating the equation \[\int \frac{2(x-1)}{x^2+4} \, dx\], we accumulate areas and curves defined by our initial differential equation. Completing the integration here requires executing the substitution, yielding a more straightforward expression \[u(x)=\int \frac{(x-1)}{t} \, dt\].
Upon simplifying, it leads us finely to a final integrative expression: \[u(x) = \frac{1}{2}\ln{\left| x^2+4 \right|} + C\]
This solution includes an arbitrary constant \(C\), ensuring that it represents a wide spectrum of answers.

• Integrating parameterizes the changes in \(x\) to assign values to \(u\).
• Recognizes both symmetry and defined value changes across intervals.