Question

\(\left(\mathrm{y}-\frac{\mathrm{xdy}}{\mathrm{dx}}\right)=\mathrm{a}\left(\mathrm{y}^{2}+\frac{\mathrm{dy}}{\mathrm{dx}}\right)\)

Short answer

Based on the given step-by-step solution, create a short answer: First, we simplify the given equation and obtain: $$\frac{dy}{dx} = \frac{ay^2 - y}{x + a}$$ Then, after separating the variables and integrating, we find the solution in the form of a quadratic equation in terms of y: $$ay^2 - y - \epsilon e^{2a(x + C)} = 0$$ where \(\epsilon = \pm1\) and \(C\) is the integration constant. We can either solve for y as a function of x using the quadratic formula, if possible, or leave the answer in this form.

Step-by-step solution

Simplify the given equation

First, let's rewrite the given equation without the parentheses to make it more manageable: $$y - \frac{xdy}{dx} = ay^2 + a\frac{dy}{dx}$$ Now, we can move all the terms involving the derivatives to one side of the equation and terms involving y to the other side of the equation: $$\frac{xdy}{dx} + a\frac{dy}{dx} = ay^2 - y$$

Separate the variables

Now we want to separate the variables so that the derivatives are on one side and the functions are on the other side. To do this, we can factor \(\frac{dy}{dx}\) out of the left side and divide both sides by the remaining terms: $$\frac{dy}{dx}(x + a) = ay^2 - y$$ $$\frac{dy}{dx} = \frac{ay^2 - y}{x + a}$$

Integrate both sides

Now we can integrate both sides of the equation with respect to x to get our solution: $$\int\frac{dy}{y^2 - \frac{y}{a}} = \int\frac{a \, dx}{x + a}$$ Let's focus on the left side integral first. We can rewrite the integral as follow: $$\int\frac{a \, dy}{ay^2 - y}$$ Now, we can make the substitution \(u = ay^2 - y, du = (2ay - 1) \, dy\). The left side integral then becomes: $$\int\frac{1}{2ay - 1}\frac{du}{u}$$ Now, we can integrate both sides: $$\frac{1}{2a}\int\frac{du}{u} = \int dx$$

Solve for y as a function of x

After integrating, we have: $$\frac{1}{2a}\ln|u| = x + C$$ Now, let's substitute u back with the expression in terms of ay: $$\frac{1}{2a}\ln|ay^2 - y| = x + C$$ To solve for y, we can take the exponential of both sides, then isolate the y terms: $$ay^2 - y = \pm e^{2a(x + C)}$$ $$ay^2 - y - \epsilon e^{2a(x + C)} = 0$$, where \(\epsilon = \pm1\) and \(C\) is the integration constant. Now, we have a quadratic equation in terms of y. Either use the quadratic formula to solve for y as a function of x, if possible, or leave the answer in this form.

Key concepts

Differential Equations

Differential equations are powerful tools in mathematics that describe relationships involving rates of change. They play a crucial role in various fields including physics, engineering, and economics. A differential equation involves an unknown function and its derivatives. The equation provided here is a classic example of a first-order differential equation:
\[y - \frac{x\,dy}{dx} = ay^2 + a\frac{dy}{dx}\]
It expresses a relationship between the rate of change of a function, denoted by \(\frac{dy}{dx}\), the function itself \(y\), and the independent variable \(x\). Solving these equations involves finding a function \(y\) that satisfies the equation for given values of \(a\) and \(x\).

Separation of Variables

Separation of variables is a method for solving differential equations, where we rearrange the equation to isolate the terms involving \(y\) on one side, and the terms involving \(x\) on the other side. This technique works well for equations where the separation is possible, as demonstrated in the solution:
\[\frac{dy}{dx}(x + a) = ay^2 - y\] becomes \[\frac{dy}{dx} = \frac{ay^2 - y}{x + a}\].
After separating the variables, we integrate both sides independently. This requires a solid grasp of integration techniques such as substitution and partial fractions, making integral calculus an indispensable part of solving these equations. The key is to arrange the differentials \(dx\) and \(dy\) on separate sides, allowing us to integrate and find the function that describes the behavior of the system in question.

Integrating Factors

In some differential equations, direct separation of variables is not possible. In those cases, integrating factors become a useful tool. An integrating factor is a function that we multiply by the differential equation to facilitate its solution. It's designed to make the left side of the equation a perfect derivative, which simplifies the process of integration. While the current exercise doesn't require an integrating factor, it's important to be aware of this technique, as it is often used when the differential equation is linear but cannot be separated. The process involves finding a function \(\mu(x)\) such that multiplying through allows for simplification and direct integration, often resulting in an easier path to find the solution to a once intractable differential equation.

Quadratic Equations

Quadratic equations are second-degree polynomials with a general form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. To solve for \(y\) in our differential equation, we eventually encounter a quadratic equation:
\[ay^2 - y - \epsilon e^{2a(x + C)} = 0\].
Solving this equation for \(y\) provides the function in terms of \(x\) that we’ve been seeking. The most common methods to solve a quadratic equation are factoring, completing the square, and using the quadratic formula. Understanding these methods allows us to handle the integral calculus solutions that result in quadratic equations in \(y\), providing specific solutions to the values that satisfy the original differential equation.