Question

Solve the given homogeneous equation implicitly. $$ y^{\prime}=\frac{x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}} $$

Short answer

To solve the given homogeneous equation implicitly, follow these steps: 1. Identify the form of the homogeneous differential equation: $$y^{\prime}=\frac{M(x,y)}{N(x,y)}$$. 2. Introduce a new dependent variable, v = y/x, and rewrite the homogeneous differential equation using this new variable. 3. Simplify the equation and make it separable. 4. Integrate the equation implicitly with respect to x. 5. Substitute v with y/x to find the implicit solution of the given homogeneous equation. The implicit solution for the given equation is: $$\frac{y}{x}+\ln|x|=\ln (x^{2}+3\left(\frac{y}{x}\right)^{2}x^{2})+C$$

Step-by-step solution

Identify the Homogeneous Differential Equation

First, we recognize that the given equation is a homogeneous differential equation of the form: $$ y^{\prime}=\frac{M(x,y)}{N(x,y)} $$ Where: $$ M(x, y) = x^3 + x^2y + 3y^3 $$ $$ N(x, y) = x^3 + 3xy^2 $$

Rewrite the Homogeneous Differential Equation with an integrating factor

Introduce a new dependent variable v = y/x. By using the chain rule, we are going to find the derivative of y in terms of x. $$ y = v*x $$ $$ y' = v' * x +v $$ Now substitute v and its derivative, v', into the given equation: $$ v' * x +v = \frac{x^{2}(x+v)+3(vx)^{2}}{x(x^{2}+3(vx)^{2})} $$

Simplify the equation

Simplify the given equation: $$ v' * x +v = \frac{x^3+x^2v+3v^2x^3}{x^3+3v^2x^3} $$ Now, divide both sides by x: $$v' + \frac{v}{x} = \frac{x^2+v+3v^2x^2}{x^2+3v^2x^2} $$

Integrate the equation implicitly

We will integrate the equation implicitly w.r.t x: $$ \int (v'+\frac{v}{x}) dx = \int \frac{x^2+v+3v^2x^2}{x^2+3v^2x^2} dx $$ Since the left side of the equation is separable, we find the integral as: $$ \int v' dx + \int \frac{v}{x} dx = \int \frac{x^2+v+3v^2x^2}{x^2+3v^2x^2} dx $$ Now integrate: $$v+\ln|x|=\ln (x^{2}+3v^{2}x^{2})+C$$ Here, C is the constant of integration.

Substitute v with y/x

Now, substitute back v = y/x: $$\frac{y}{x}+\ln|x|=\ln (x^{2}+3\left(\frac{y}{x}\right)^{2}x^{2})+C$$ This is the solution to the given homogeneous equation implicitly.

Key concepts

Implicit Integration

Implicit integration involves solving equations without isolating the dependent variable explicitly. Instead, we work with relationships between variables to find a solution that satisfies the differential equation. In the context of the given homogeneous differential equation, we start by rearranging and simplifying the equation to facilitate integration.
Implicit integration helps us find a solution that connects the variables naturally, often expressed as a relationship form with an integration constant. Here, after simplifying and substituting, we integrate:

• The left side becomes: \(\int (v'+\frac{v}{x}) \, dx\)
• The right side is simplified to: \(\int \frac{x^2+v+3v^2x^2}{x^2+3v^2x^2} \, dx\)

This method allows us to integrate both sides and arrive at the solution without needing to change the form of the solution statement drastically, maintaining an implicit relationship between x and y.

Substitution Method

The substitution method is a powerful tool in solving differential equations, particularly homogeneous ones. In our equation, we utilize a transformation, specifically substituting \(v = \frac{y}{x}\). This allows us to simplify the differential equation, making it easier to manipulate and solve.
When we perform this substitution:

• \(y = v \cdot x\)
• \(y' = v' \cdot x + v\)

The substitution helps us convert the original complex expressions into a form that can be easily managed with standard calculus operations. This process not only simplifies integration but also aids in identifying symmetries or patterns within the equation.
By reformulating in terms of v and its derivative, the equation aligns more closely with patterns suitable for separation of variables or integrating factors, facilitating further steps toward integration.

Separation of Variables

Separation of variables is a technique used to rearrange a differential equation so that each variable and its differential are on opposite sides of the equation. For the given equation, after making the substitution, we end up with:

• The left-hand side: \(v' + \frac{v}{x}\)
• The right-hand side: \(\frac{x^2+v+3v^2x^2}{x^2+3v^2x^2}\)

By simplifying and rearranging, we place differentials of v and x on separate sides. This makes it possible to integrate with respect to x on both sides.
Separation of variables is effective for many differential equations, especially when direct integration isn't straightforward. Even if the equation does not initially appear to be separable, substitutions like \(v = \frac{y}{x}\) can transform it into a manageable form, allowing us to apply calculus effectively to find a solution.

Integrating Factor

An integrating factor is a function that simplifies a differential equation into a form that can be more easily integrated. Although not explicitly used in this solution, understanding integrating factors gives context to techniques used on similar equations. When a differential equation is not easily integrable, a function multiplied throughout the equation can make it exact or easier to solve.
For instance, in linear differential equations of the form \(y' + P(x)y = Q(x)\), an integrating factor, typically \(e^{\int P(x) \, dx}\), transforms the left side into an exact derivative \(\frac{d}{dx}( I(x) y)\). Though our example utilizes implicit integration, acknowledging integrating factors emphasizes the diversity of methods available for simplifying and solving differential equations.
By recognizing when and how to apply different techniques, such as integrating factors, we can tackle a wide array of problems even when they don't immediately seem approachable with basic calculus tools.