Question

Find an integrating factor and solve the given equation. $$ \left(3 x+\frac{6}{y}\right)+\left(\frac{x^{2}}{y}+3 \frac{y}{x}\right) \frac{d y}{d x}=0 $$

Short answer

Question: Solve the given first-order ordinary differential equation (ODE) using an integrating factor. $$ \left(3x +\frac{6}{y}\right) +\left(\frac{x^{2}}{y}+3 \frac{y}{x}\right) \frac{d y}{d x}=0 $$ Answer: The equation cannot be solved using a simple closed-form solution due to the complexity of the integral. However, alternative methods such as numerical integration can be used to approximate a solution.

Step-by-step solution

Rewrite in standard form

The first step is to rearrange the given ODE to have the standard linear ODE form as mentioned above. $$ \left(\frac{x^{2}}{y}+3 \frac{y}{x}\right) \frac{d y}{d x} = -\left(3x +\frac{6}{y}\right) $$ Substitute \(P(x, y) = \frac{x^{2}}{y} + 3 \frac{y}{x}\) and \(Q(x, y) = -\left(3x +\frac{6}{y}\right)\):

Calculate the integrating factor

To find the integrating factor, take the derivative of \(P(x, y)\) with respect to \(x\). \(\frac{dP}{dx} = \frac{d}{dx} \left(\frac{x^{2}}{y} + 3 \frac{y}{x}\right) = \frac{2x}{y} - 3 \frac{y}{x^2}\) Next, divide \(\frac{dP}{dx}\) by \(P(x, y)\): \(\frac{\frac{dP}{dx}}{P(x, y)} = \frac{\frac{2x}{y} - 3 \frac{y}{x^{2}}}{\frac{x^{2}}{y} + 3 \frac{y}{x}}\) Now, we integrate the result with respect to \(x\) to find the integrating factor: \(\int \frac{\frac{dP}{dx}}{P(x, y)} dx = \int \frac{\frac{2x}{y} - 3 \frac{y}{x^{2}}}{\frac{x^{2}}{y} + 3 \frac{y}{x}} dx\) Noticing that the numerator and denominator are equal, the integral becomes: \(\int 1 dx = x\) So, the integrating factor is \(IF(x) = x\).

Apply the integrating factor

Now, you need to multiply the ODE by the integrating factor x: $$ x\left(\left(\frac{x^{2}}{y}+3 \frac{y}{x}\right) \frac{d y}{d x}\right) = x\left(-\left(3x +\frac{6}{y}\right)\right) $$ Simplify the equation: $$ \left(x^2 + 3y^2\right) \frac{dy}{dx} = -\left(3x^2 + \frac{6x}{y}\right) $$

Integrate the equation

Now, we need to integrate both sides of the equation. First, rearrange it as: $$ \frac{dy}{dx} = -\frac{3x^2 + \frac{6x}{y}}{x^2 + 3y^2} $$ Now, integrate both sides: $$ \int dy = \int -\frac{3x^2 + \frac{6x}{y}}{x^2 + 3y^2} dx $$ Unfortunately, this integral is quite complex, and there is no simple closed-form solution. Nonetheless, you can still try alternative methods, such as numerical integration, to get an approximate solution.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that relate a function with its derivatives. They're called "ordinary" because they involve a single variable function, unlike partial differential equations that deal with multiple variables. ODEs are fundamental in describing various phenomena, including physics, engineering, and biology.

For example, Newton's second law, which describes the motion of a particle under a force, can be formulated as an ordinary differential equation. Solving ODEs typically involves finding a function that satisfies the equation. This process often requires unique approaches based on the type of functions and derivatives involved in the problem.

Linear ODE

Linear Ordinary Differential Equations are a special class of ODEs where the dependent variable and all its derivatives appear linearly. They follow the principle of superposition and have solutions that are particularly well-behaved, that is, they can be expressed in terms of exponential, sinusoidal, and polynomial functions. The general form of a linear ODE is given by:

\[ a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \, ... \, + a_1(x)y' + a_0(x)y = f(x) \]

In this form, the coefficients \(a_n(x), a_{n-1}(x), ..., a_0(x)\) are functions of the independent variable, and \(f(x)\) is a known function. Linear ODEs are easier to solve compared to nonlinear ones, and their solutions often involve techniques like integrating factors, undetermined coefficients, or variation of parameters.

Exact Differential Equations

Exact Differential Equations are a subset of ODEs where the equation can be expressed as a total derivative of some function. They generally take the form:

\[ M(x, y)dx + N(x, y)dy = 0 \]

Such an equation is considered exact if there exists a function \( \psi(x, y)\) such that \( \frac{\partial \psi}{\partial x} = M \) and \( \frac{\partial \psi}{\partial y} = N \). If this condition holds, the solution can be directly written as \( \psi(x, y) = C \), where \( C \) is a constant.

However, not all differential equations are exact. If an ODE is not exact, sometimes it can be made into an exact ODE using an integrating factor. This is an important transformative tool, which is particularly useful when direct integration isn't feasible.

Numerical Integration

Numerical Integration is a technique employed when an analytic solution to a differential equation is difficult or impossible to find. It involves approximating the solution by discretizing the problem and solving it at specific points. This is particularly useful for non-linear or complex equations that don't have solutions in closed form.

Some common numerical methods include:

• Euler's Method: A simple one-step method that's easy to implement but may be less accurate for small step sizes.
• Runge-Kutta Methods: A family of iterative methods that provide higher accuracy, particularly popular is the fourth-order Runge-Kutta method.
• Finite Difference Method: Used to approximate solutions by replacing derivatives with difference equations.

Numerical integration enables solving real-world problems that are analytically intractable, making it essential in fields like engineering and physics.