Chapter 30: Problem 7
Find the general solution to each differential equation. $$y^{\prime}=\frac{3-x y}{2 x^{2}}$$
Short Answer
Expert verified
The general solution to the differential equation is \( y = \frac{3x+C}{x^2} \), where C is the constant of integration.
Step by step solution
01
Identify Type of Differential Equation
Recognize that the equation is a first-order non-linear differential equation. An approach to this is separating the variables if possible.
02
Separate the Variables
Rewrite the equation to separate the variables x and y. Move all terms containing y to one side and those containing x to the other side: \( y' = \frac{3 - xy}{2x^2} \Rightarrow 2x^2 y' = 3 - xy \Rightarrow y + 2x^2 y' = 3 \Rightarrow y + 2x^2 \frac{dy}{dx} = 3 \Rightarrow \frac{dy}{dx} + \frac{y}{2x^2} = \frac{3}{2x^2} \).
03
Adjust Terms for Integration
To integrate, express the left side such that it becomes a derivative of a product: \( \frac{d}{dx}(yx^2) = 3 \). This step comes from noticing that \( y + 2x^2y' \) is the derivative of \( yx^2 \) with respect to x.
04
Integrate Both Sides
Integrate both sides with respect to x: \( \int \frac{d}{dx}(yx^2)dx = \int 3dx \), which results in \( yx^2 = 3x + C \) where C is the constant of integration.
05
Solve for y
Divide both sides by \( x^2 \) to solve for y: \( y = \frac{3x+C}{x^2} \). This is the general solution to the differential equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
First-Order Non-Linear Differential Equation
When faced with a first-order non-linear differential equation, it is essential to recognize the characteristics that set it apart from its linear counterparts. A linear equation maintains proportionality and additive constants within its expressions, while a non-linear equation, similar to the given exercise \(y' = \frac{3 - xy}{2x^2}\), contains variables that are either multiplied together or involve exponents other than one.
Delving into the math, linear equations form a straight line on a graph, and solutions to linear differential equations involve exponential functions. Conversely, non-linear equations can create a variety of curves, making their solutions more complex and diverse, often requiring specialized methods to solve.
In our example, the presence of an xy term indicates a non-linear relationship between x and y. The method we used to solve the equation is separating variables, which isn't typically applicable to non-linear equations unless they can be manipulated into a separable form. This type of equation offers a rich field of study due to its intricate behavior and the complexity of its solutions.
Delving into the math, linear equations form a straight line on a graph, and solutions to linear differential equations involve exponential functions. Conversely, non-linear equations can create a variety of curves, making their solutions more complex and diverse, often requiring specialized methods to solve.
In our example, the presence of an xy term indicates a non-linear relationship between x and y. The method we used to solve the equation is separating variables, which isn't typically applicable to non-linear equations unless they can be manipulated into a separable form. This type of equation offers a rich field of study due to its intricate behavior and the complexity of its solutions.
Separating Variables
The method of separating variables is a foundational technique for solving differential equations that involve two variables multiplicatively linked. In this method, each variable and its differential are moved to opposite sides of the equation. This technique facilitates the integration process by creating integrals that are only functions of a single variable.
Applying this to our exercise, we have for example \(y + 2x^2 y' = 3\). With skillful manipulation, such as recognizing that terms involving y can be written as the derivative of a product involving y and x, we can express the equation in a form that allows us to separate the variables. However, as is typical with non-linear equations like ours, specifically arranging terms to enable separation can be challenging and requires insight into the equation's structure.
The power of this technique lies in its simplicity and the intuitive appeal of integrating functions of one variable at a time, allowing for most first-order ordinary differential equations that are separable to be resolved directly through integration.
Applying this to our exercise, we have for example \(y + 2x^2 y' = 3\). With skillful manipulation, such as recognizing that terms involving y can be written as the derivative of a product involving y and x, we can express the equation in a form that allows us to separate the variables. However, as is typical with non-linear equations like ours, specifically arranging terms to enable separation can be challenging and requires insight into the equation's structure.
The power of this technique lies in its simplicity and the intuitive appeal of integrating functions of one variable at a time, allowing for most first-order ordinary differential equations that are separable to be resolved directly through integration.
Integrating Factors
Integrating factors are a potent tool that can transform certain non-linear differential equations into forms that allow for direct integration. The use of integrating factors is a technique that involves multiplying the entire differential equation by a strategically chosen function, enabling the left-hand side of the equation to be expressed as the derivative of a product of two functions.
In the example given, while the method of separating variables was effective, in more challenging situations where separating variables isn't possible, an integrating factor would be needed. An integrating factor is typically a function of x only or y only, derived from the equation itself. This function, when found, can turn an intractable equation into one that conforms to the methods of direct integration.
The selection of the right integrating factor is pivotal, which often depends on recognizing a pattern in the differential equation that suggests a function whose derivative is present within the equation. Once the equation is multiplied by this integrating factor, the left side of the equation can often be written as the derivative of a product, yielding a form that can be easily integrated.
In the example given, while the method of separating variables was effective, in more challenging situations where separating variables isn't possible, an integrating factor would be needed. An integrating factor is typically a function of x only or y only, derived from the equation itself. This function, when found, can turn an intractable equation into one that conforms to the methods of direct integration.
The selection of the right integrating factor is pivotal, which often depends on recognizing a pattern in the differential equation that suggests a function whose derivative is present within the equation. Once the equation is multiplied by this integrating factor, the left side of the equation can often be written as the derivative of a product, yielding a form that can be easily integrated.
