Question

Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. $$x^{2} y^{\prime}(x)=y^{2}, x>0$$

Short answer

Answer: The general solution for the given first-order differential equation is \(y(x) = \frac{x}{1 - Cx}\).

Step-by-step solution

Rewrite the differential equation

Given the differential equation: $$x^{2} y'(x) = y^{2}, x > 0$$ Rewrite it using the derivative notation: $$x^{2} \frac{dy}{dx} = y^{2}$$

Separate the variables

Divide both sides of the equation by \(y^2\) and by \(x^2\) to isolate the y terms and the x terms: $$\frac{1}{y^2} \frac{dy}{dx} = \frac{1}{x^2}$$

Integrate both sides

Now, integrate both sides with respect to x: $$\int \frac{1}{y^2} \frac{dy}{dx} dx = \int \frac{1}{x^2} dx$$ The left side of the equation can be rewritten as a single integral with respect to y: $$\int \frac{1}{y^2} dy = \int \frac{1}{x^2} dx$$ Now, we compute the integrals: $$-\frac{1}{y} = -\frac{1}{x} + C$$

Solve for y

Let's multiply both sides by \(-1\) to make the equation simpler: $$\frac{1}{y} = \frac{1}{x} - C$$ Now, take the reciprocal of both sides to solve for y and express the general solution explicitly as a function of the independent variable x: $$y(x) = \frac{1}{\frac{1}{x} - C} = \frac{x}{1 - Cx}$$ The general solution of the differential equation is: $$y(x) = \frac{x}{1 - Cx}$$

Key concepts

Separable Differential Equations

Differential equations are equations involving derivatives of a function or functions. A separable differential equation is a type that can be rewritten in a way that separates all terms involving the dependent variable from those involving the independent variable. In essence, we arrange the equation so that each side of the equation contains only one variable. This makes it simple to integrate both sides.

• For example, starting with an equation like \( x^2 \frac{dy}{dx} = y^2 \), we aim to separate it into \( \frac{dy}{y^2} = \frac{dx}{x^2} \).
• This separation allows each variable to be integrated independently, leading to a solution.

Separable differential equations are foundational because they introduce basic techniques for solving more complex differential equations. Recognizing the equation's form allows for straightforward variable separation and subsequent integration.

Integration

Integration is a crucial mathematical process used in calculus to find the antiderivative of functions. In the context of differential equations, integration helps us find functions whose derivatives satisfy the given equation. After separating the variables in our differential equation, each side can be integrated independently:

• For the separated form \( \int \frac{1}{y^2} dy = \int \frac{1}{x^2} dx \), integration lets us deduce the relationship between variables.
• The left side is solved by recognizing \( \int y^{-2} dy = -y^{-1} = -\frac{1}{y} \).
• Similarly, the right side finds its antiderivative as \( \int x^{-2} dx = -x^{-1} = -\frac{1}{x} \).

After computing each integral, we combine them, often introducing a constant of integration. This constant, \( C \), signifies the family of possible solutions. Thus, the integrals emerge as \( -\frac{1}{y} = -\frac{1}{x} + C \).

General Solution

The general solution of a differential equation comprises all possible solutions. It has one or more arbitrary constants, reflecting its ability to model various scenarios. By computing the integrals from the previous step, we find:

• After solving for the dependent variable, \( y \), we obtain \( \frac{1}{y} = \frac{1}{x} - C \).
• To express \( y \) clearly: multiply through by \(-1\) and solve to yield \( y = \frac{x}{1 - Cx} \).

This expression represents the general solution: an equation that describes an entire family of curves on a graph. Each variation in the constant \( C \) results in a different curve. This family of solutions provides great flexibility in modeling real-world scenarios.

Variable Separation

Variable separation is a method employed in solving separable differential equations by placing all terms of one variable on one side of the equation, and all terms with the other variable on the opposite side. It's a straightforward and powerful tool, especially for differential equations fitting this form.

• The main aim is to simplify the differential equation into two integrable sides: \( \frac{1}{y^2} dy = \frac{1}{x^2} dx \).
• This approach allows easy integration as each variable is isolated and solvable independently.
• The process simplifies finding the antiderivative and hence the solution to the differential equation.

Variable separation is often among the first techniques taught in differential equations, emphasizing the importance of recognizing when a differential equation can be manipulated into a separable form.