Question

Find the "general solution" (that is, a solution containing an arbitrary constant) of each of the following differential equations, by separation of variables. Then find a particular solution of each equation satisfying the given boundary conditions. \(y^{\prime}=\frac{2 x y^{2}+x}{x^{2} y-y}\) \(y=0\) when \(x=\sqrt{2}\)

Short answer

The general solution is \( y^2/2 = x^3/3 - x + C \) and the particular solution is \( y = 2\sqrt{(x^3/3) - x + 2/3} \)

Step-by-step solution

Rewrite the differential equation

Express the given differential equation in a separable form. The original equation is: \( y^{\prime} = \frac{2xy^2 + x}{x^2y - y} \)

Simplify the equation

Factor the common terms in both the numerator and the denominator: \( y^{\prime} = \frac{x(2y^2 + 1)}{y(x^2 - 1)} \)

Separate the variables

Separate the variables by grouping all terms involving \(y\) on one side and \(x\) on the other side: \( y \cdot (x^2 - 1) dy = x(2y^2 + 1) dx \)

Integrate both sides

Integrate both sides of the equation: \( \int y \cdot (x^2 - 1) dy = \int x(2y^2 + 1) dx \)

Solve the integral

Solve the integral on both sides. Let's integrate: For the left side: \( \int y dy = \frac{y^2}{2} \) For the right side: \( x^3/3 - x \)

Obtain the general solution

Combine and simplify the integrated forms for the general solution: \( y^2/2 = x^3/3 - x + C \)

Apply the boundary condition

Use the boundary condition \( y = 0 \) when \( x = \sqrt{2} \) to find the constant \(C\). Substitute values into the general solution: \( 0 = 2/3 - x + C \) Solving for \(C\): \( C = 0 C = 2/3 \)

Obtain the particular solution

Substitute the value of \(C\) into the general solution to get the particular solution: \( y^2/2 = x^3/3 - x + 2/3 \) Simplify: \( y = 2\sqrt{(x^3/3) - x + 2/3} \)

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. These are crucial tools in modeling real-world phenomena where rates of change are involved, such as in physics, engineering, and economics. In this exercise, we are dealing with a first-order differential equation. A first-order differential equation involves the first derivative of the function, denoted as \(y'\). To solve such equations, we need to find the function \(y\) that satisfies the given relationship with its derivative. This process often involves techniques like separation of variables, as demonstrated below.

Boundary Conditions

Boundary conditions are specific values for the function or its derivatives at particular points, which help in finding a unique solution to a differential equation. Given the boundary condition \(y = 0\) when \(x = \sqrt{2}\), we use these values to find the arbitrary constant in the general solution. Boundary conditions basically 'anchor' the solution, ensuring it aligns with specific constraints or real-world measurements.

Integration Techniques

Integration techniques involve finding antiderivatives of functions. In this exercise, we encountered the need to integrate both sides of the differential equation. The left side required integrating \(y \cdot (x^2 - 1) dy\), and the right side required integrating \(x \cdot (2y^2 + 1) dx\). In general, successful integration often involves:

• Simplifying the integrand, the function to be integrated
• Identifying proper integration techniques, such as substitution or integration by parts
• Applying known antiderivative formulas

After integrating both sides, we obtained the form that leads us to our general solution.

General and Particular Solutions

A general solution of a differential equation includes all possible solutions and has an arbitrary constant, noted as \(C\). In our exercise, the general solution is:
\[ \frac{y^2}{2} = \frac{x^3}{3} - x + C \]
To find the particular solution, we apply the given boundary conditions. By substituting \(y = 0\) and \(x = \sqrt{2}\), we solve for \(C\). The particular solution then eliminates the arbitrary constant, providing a specific function that exactly satisfies the conditions. Thus, this specific solution aligns perfectly with the problem's constraints, modeling the exact scenario set by the boundary conditions.