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Magazine Chapter 9: Problem 3
Solve the differential equations. \(x y^{\prime}+3 y=\frac{\sin x}{x^{2}}, \quad x>0\)
Short Answer
Expert verified
The solution is \(y = \frac{-\cos x + C}{x^3}\).
Step by step solution
01
Identify the Equation Type
The given equation is a first-order linear differential equation of the form \(xy' + 3y = \frac{\sin x}{x^2}\). We can identify this by noticing that it fits the structure \(P(x)y' + Q(x)y = R(x)\), where \(P(x) = x\), \(Q(x) = 3\), and \(R(x) = \frac{\sin x}{x^2}\).
02
Write in Standard Form
Divide the entire differential equation by \(x\) to get the standard form: \[ y' + \frac{3}{x}y = \frac{\sin x}{x^3}, \quad x > 0. \]
03
Identify Integrating Factor
The integrating factor \(\mu(x)\) is given by \(e^{\int P(x)\,dx}\), where \(P(x) = \frac{3}{x}\). Calculate the integrating factor: \[ \mu(x) = e^{\int \frac{3}{x}\,dx} = e^{3\ln x} = x^3. \]
04
Multiply Equation by Integrating Factor
Multiply both sides of the standard form by the integrating factor \(x^3\): \[ x^3y' + 3x^2y = \sin x. \]
05
Convert to Exact Differential
Now, the left side is the derivative of a product: \(\frac{d}{dx}(x^3y) = \sin x\). This converts our equation to: \[ \frac{d}{dx}(x^3y) = \sin x. \]
06
Integrate Both Sides
Integrate both sides with respect to \(x\): \[ x^3y = \int \sin x \, dx = -\cos x + C, \] where \(C\) is the constant of integration.
07
Solve for y
Solve the equation for \(y\): \[ y = \frac{-\cos x + C}{x^3}. \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integrating Factor
In first-order linear differential equations, an integrating factor is a crucial tool that helps us solve the equation. It is a function used to multiply through the entire differential equation to simplify it into an exact differential form.
The general strategy involves transforming the differential equation by finding this factor for the equation to become more manageable. To determine the integrating factor \(\mu(x)\), we usually use the formula:
In the example provided, with \(P(x) = \frac{3}{x}\), the integrating factor becomes \(\mu(x) = e^{3 \ln x} = x^3\).
Applying this factor transforms the equation in a way that makes integration feasible through simplification, a key to solving the differential equation.
The general strategy involves transforming the differential equation by finding this factor for the equation to become more manageable. To determine the integrating factor \(\mu(x)\), we usually use the formula:
- \( \mu(x) = e^{\int P(x) \,dx} \)
In the example provided, with \(P(x) = \frac{3}{x}\), the integrating factor becomes \(\mu(x) = e^{3 \ln x} = x^3\).
Applying this factor transforms the equation in a way that makes integration feasible through simplification, a key to solving the differential equation.
Exact Differential
An exact differential represents a differential equation where integration can be performed directly. This occurs when the left-hand side of the equation after multiplying by the integrating factor is the derivative of a function.
When we successfully find an integrating factor, our goal is to cast the left side of the equation in this form:
This indicates that the differential expression has been made exact, and we can now focus on integrating the right side with respect to \(x\), streamlining the solution process.
When we successfully find an integrating factor, our goal is to cast the left side of the equation in this form:
- \( \frac{d}{dx}(g(x)y) = h(x) \)
This indicates that the differential expression has been made exact, and we can now focus on integrating the right side with respect to \(x\), streamlining the solution process.
Standard Form of Differential Equations
To solve a linear first-order differential equation effectively, it's often best to start by writing it in standard form. The standard form is of the structure:
For the given problem, we initially had the equation \(xy' + 3y = \frac{\sin x}{x^2}\).
By dividing the entire equation by \(x\), we reformulated it into the standard form: \( y' + \frac{3}{x}y = \frac{\sin x}{x^3}\).
Recognizing this form is vital as it sets the stage for applying the method of integrating factors and ultimately guides us through transforming and solving the differential equation more smoothly.
- \( y' + P(x)y = Q(x) \)
For the given problem, we initially had the equation \(xy' + 3y = \frac{\sin x}{x^2}\).
By dividing the entire equation by \(x\), we reformulated it into the standard form: \( y' + \frac{3}{x}y = \frac{\sin x}{x^3}\).
Recognizing this form is vital as it sets the stage for applying the method of integrating factors and ultimately guides us through transforming and solving the differential equation more smoothly.
