Question

Solution of \(\left(x^{2}+y^{2}\right) d y=x y d x\) is

Short answer

The solution is \(y = Cx\), a family of lines through the origin.

Step-by-step solution

Rewrite the Given Equation

The given equation is \((x^2 + y^2) dy = xy dx\). To make it more convenient for solving, rewrite it in the form of a differential equation: \(\frac{dy}{dx} = \frac{xy}{x^2+y^2}\).

Separate Variables

Aim to separate variables so that all \(y\) terms are on one side and all \(x\) terms are on the other. We separate variables to get \( (x^2 + y^2)dy = xy dx \). Moving terms around, this equation cannot be solved by separation of variables directly, so we'll use another method.

Recognize Function Form

The equation is a homogeneous differential equation since the degree of numerator and denominator terms in \(\frac{dy}{dx}\) is the same. Recognize that substitution can simplify the equation.

Use Substitution Method

Use the substitution \(v = \frac{y}{x}\), hence \(y = vx\) and \(dy = v dx + x dv\). Substitute these into the differential equation.

Substitute and Simplify

Substitute \(y = vx\) and \(dy = v dx + x dv\) into \((x^2 + y^2) dy = xy dx\) to get \(x^2(1 + v^2)(v dx + x dv) = x^2v dx\). Simplify to \((1 + v^2)x dv = 0\).

Solve the Simplified Equation

The equation simplifies to \((1 + v^2) dv = 0\). Solving this equation by integration gives \(\int (1+v^2) dv = 0\) leading to \(v = 0 + C\).

Rewrite in Terms of y and x

Substitute \(v\) back as \(v = \frac{y}{x}\). Therefore, \(\frac{y}{x} = C\) or equivalently \(y = Cx\).

General Solution

The general solution form \(y = Cx\) describes a family of straight lines through the origin where \(C\) is a constant. This is the implicit solution to the differential equation.

Key concepts

Homogeneous differential equations

In the study of differential equations, one important class is known as homogeneous differential equations. These are equations where the degrees of every term in the numerator and denominator are the same. This gives them a special property: they can often be simplified through substitution. Homogeneous equations usually look complex at first glance because they involve expressions like polynomials, but they have structured ways to be approached. To identify a homogenous differential equation, you divide the variables of the equation and check if each term holds the same total degree when examined. For example, in the equation \( \frac{dy}{dx} = \frac{xy}{x^2+y^2} \), both numerator and denominator contain terms of degree 2. Recognizing this property is key to finding the solution.

Substitution method

The substitution method is a powerful tool used to simplify and solve homogeneous differential equations. When an equation can be characterized as homogeneous, we can use a particular substitution to reduce its complexity. Usually, the substitution \( v = \frac{y}{x} \) is applied. This allows us to express \( y \) in terms of \( x \) and the new variable \( v \), so \( y = vx \). The differential \( dy \) becomes \( v dx + x dv \), making it easier to plug back into the differential equation.By doing this, we change the variables in the equation and often eliminate one of them, transforming the equation into a simpler format that can be solved more easily.

Separation of variables

Separation of variables is another critical method for solving differential equations. It involves manipulating an equation to have all terms involving one variable on one side of the equation and all terms involving the other variable on the other side. This setup allows for straightforward integration on both sides. In the given example, initially, an attempt to separate the variables directly was not fruitful because the equation wasn't amenable to this method in its original form. The equation was rewritten and then classified under homogeneous equations which led us to use the substitution method instead. This highlights that while separation of variables is powerful, it is sometimes not directly applicable, and recognizing when an alternative method is needed is crucial.

Integration

Integration is the process of finding the antiderivative of a function, and it's a necessary step in solving differential equations. After successfully performing substitution or separating variables, integrating is typically the next step to finding the solution of the transformed equation.For the equation \( (1 + v^2) dv = 0 \), integration straightforwardly gives \( \int (1+v^2) dv = 0 \). Solving this integral, we find \( v = C \), where \( C \) is an integration constant. This process is essential in moving from a differential equation to a functional equation that can describe a general solution.

General solution

The general solution is the ultimate goal when solving a differential equation. It gives us a set of possible solutions that depend on arbitrary constants derived from integration. These solutions often describe families of curves or lines that satisfy the original differential equation.In our case, after substituting back \( v = \frac{y}{x} \), we get \( \frac{y}{x} = C \) or equivalently \( y = Cx \). This represents a family of straight lines through the origin. Here, \( C \) is an arbitrary constant that can be adjusted to fit specific initial conditions if provided. Understanding general solutions is crucial for interpreting how different functions can satisfy the conditions of a differential equation while retaining a parameter that allows flexibility.