Problem 34 - Calculus Early Transcendentals

Question

Find the general solution of the following equations. $$x^{2} \frac{d w}{d x}=\sqrt{w}(3 x+1)$$

Step-by-Step Solution

Answer

A: The general solution is \(w(x)=3\ln{|x|}-\frac{1}{x}+C\).

Step 1: Separate the variables

Rewrite the given equation in the form of \(\frac{d w}{\sqrt{w}}=\frac{(3x+1)}{x^2}dx\) $$\frac{d w}{\sqrt{w}}=\frac{(3x+1)}{x^2}dx$$

Step 2: Integrate both sides

Integrate with respect to their respective variables: $$\int \frac{d w}{\sqrt{w}}=\int \frac{(3x+1)}{x^2}dx$$

Step 3: Solve the left side integral

To integrate the left side, perform a substitution (\(u=w^{1/2}\)): $$\int \frac{d w}{\sqrt{w}}=2\int u \, d u =u^2+C_{1}=w(x)$$

Step 4: Solve the right side integral

To integrate the right side, divide the expression into two separate fractions and integrate each term: $$\int \frac{(3x+1)}{x^2}dx = 3\int \frac{1}{x}dx+\int \frac{1}{x^2}dx\\\implies 3\ln{|x|}-\frac{1}{x}+C_{2}$$

Step 5: Equate both sides and solve for w(x)

Now equate both integrals and isolate w(x): $$w(x)=3\ln{|x|}-\frac{1}{x}+C$$ Where \(C=C_{2}-C_{1}\) is the constant of integration. Thus, the general solution of the given differential equation is: $$w(x)=3\ln{|x|}-\frac{1}{x}+C$$

Key Concepts

Separation of Variables

Separation of Variables is a technique used to solve differential equations. It involves rearranging the equation so that each type of variable is isolated on either side of the equation. This is a fundamental skill in solving differential equations effectively. In the given problem, we start with the equation:

• \( x^{2} \frac{d w}{d x} = \sqrt{w}(3x+1) \)

To separate the variables, rewrite the equation to isolate \( w \) terms on one side and \( x \) terms on the other. This leads to:

• \( \frac{d w}{\sqrt{w}} = \frac{(3x+1)}{x^2} dx \)

This setup allows each side to be integrated with respect to its respective variable, making it much easier to find the solution to the differential equation.

Integration Techniques

Integration is a crucial part of solving differential equations - once variables are separated. We then integrate both sides to find a general solution.

Integrating the Left Side

To integrate the left side \( \int \frac{d w}{\sqrt{w}} \), we perform a substitution. Let \( u = w^{1/2} \), which simplifies the integral to:

• \( 2 \int u \, du \), and solving yields \( u^2 + C_1 = w(x) \)

Integrating the Right Side

When integrating the right side \( \int \frac{(3x+1)}{x^2}dx \), split it into two fractions:

• \( 3 \int \frac{1}{x}dx + \int \frac{1}{x^2}dx \), resulting in \( 3\ln{|x|} - \frac{1}{x} + C_2 \)

It’s important to know basic integration techniques and substitutions to handle different types of integrals.

General Solution

The General Solution is the expression that represents all potential solutions of a differential equation. It includes a constant of integration which adjusts to meet any initial conditions.After integrating both sides, we obtain two expressions:

• From the left, \( u^2 = w(x) \)
• From the right, \( 3\ln{|x|} - \frac{1}{x} + C_2 \)

We equate them:

• \( w(x) = 3\ln{|x|} - \frac{1}{x} + C \)

The constant \( C \) represents all possible constants of integration \( C_2 - C_1 \). Thus, this expression represents the general solution that satisfies the differential equation for any given initial value. Understanding how to form and interpret the general solution is critical for solving any differential equation thoroughly.