Problem 35 - Thomas Calculus

Question

This last equation is linear in the (unknown) dependent variable u. Solve the differential equations. \(x y^{\prime}+y=y^{-2}\)

Step-by-Step Solution

Answer

The solution is \( y = \left(\frac{|C|}{x}\right)^{1/3} \).

Step 1: Identify the Equation Type

The given equation is \( x y' + y = y^{-2} \), which is a first-order differential equation because it involves the first derivative of \( y \). We can solve this using an appropriate method for first-order equations.

Step 2: Rearrange Equation

Rearrange the equation to express it in standard linear form. The equation \( x y' + y = y^{-2} \) can be written as \( y' + \frac{1}{x} y = \frac{1}{x y^2} \). This is now in the standard linear form \( y' + P(x)y = Q(x) \) where \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{xy^2} \).

Step 3: Determine Integrating Factor

For a linear first-order differential equation, we find an integrating factor \( \mu(x) \) using \( \mu(x) = e^{\int P(x) \, dx} \). Here, \( P(x) = \frac{1}{x} \), so \( \mu(x) = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \).

Step 4: Solve Using Integrating Factor

Multiply the entire differential equation by the integrating factor \( |x| \):\[ |x| y' + |x|^2 y = \frac{|x|}{x y^2} \]. This simplifies to \( (|x|y)' = \frac{1}{y^2} \).

Step 5: Integrate Both Sides

Integrate both sides with respect to \( x \). The left side becomes \( |x| y = \int \frac{1}{y^2} \, dx \), which becomes \( |x| y = -\frac{1}{y} + C \) after integrating \( -\frac{1}{y} \).

Step 6: Solve for y

Rearrange the integrated equation to solve for \( y \). We obtain \( |x| y^3 = -1 + Cy^3 \). This implies \( y^3 = \frac{|C|}{x} \). Take the cube root to find \( y \): \( y = \left(\frac{|C|}{x}\right)^{1/3} \).

Key Concepts

Integrating Factor

An integrating factor is a crucial component when solving linear first-order differential equations. It transforms these equations into a form that is easier to integrate and solve. To find the integrating factor, denoted as \( \mu(x) \), we use the expression \( \mu(x) = e^{\int P(x) \, dx} \), where \( P(x) \) is the coefficient function of \( y \) in the equation.

In our exercise, the differential equation is already in linear form:

• \( y' + \frac{1}{x} y = \frac{1}{xy^2} \)
• The function \( P(x) = \frac{1}{x} \) is identified here.

The next step is to determine the integrating factor:

• Calculate \( \mu(x) = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \).

This integrating factor \( |x| \) enables us to rewrite the differential equation, making it integrable and leading us to the solution more effectively.

Linear Differential Equation

A linear differential equation is characterized by its order and linearity over the unknown function and its derivatives. In our context, the equation is both a first-order and linear differential equation because it involves only the first derivative of \( y \), without any exponents or multiplications among the derivatives and the solution itself.

To identify a linear differential equation, it follows the general form:

• \( y' + P(x) y = Q(x) \)

In this exercise, this form is represented as:

• \( y' + \frac{1}{x} y = \frac{1}{x y^2} \)
• \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{x y^2} \)

By recognizing and rearranging the equation into this standard form, we're able to apply the method of integrating factors to find a solution efficiently.

Solving Differential Equations

Solving differential equations involves several strategic steps that once mastered, simplify seemingly complicated equations.

For our linear first-order differential equation, after identifying the equation form and determining the integrating factor, the next step is to solve it using the integrating factor:

• Multiplied through by \( |x| \), the equation becomes \( (|x| y)' = \frac{1}{y^2} \).
• Integrate both sides with respect to \( x \).

This integration provides a direct path to the general solution. The encoded equation transforms to:

• Integrate to get \( |x| y = -\frac{1}{y} + C \)
• Rearrange it to \( |x| y^3 = -1 + Cy^3 \).

The final solution for \( y \) is derived by isolating \( y \):

• Extract from the cube: \( y = \left(\frac{|C|}{x}\right)^{1/3} \)

This step-by-step approach, along with understanding of linearity and integrating factors, allows us to solve these differential equations systematically.