Problem 48
Question
In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x^{3}}-x^{3}, x>0 $$
Step-by-Step Solution
Verified Answer
The general solution is \( y(x) = -\frac{1}{x^2} - \frac{x^4}{4} + C \).
1Step 1: Identify the Type of Differential Equation
The given differential equation is \( \frac{d y}{d x} = \frac{2}{x^3} - x^3 \). It is a first-order ordinary differential equation that can be solved using the method of integration.
2Step 2: Integrate the Equation
To find \( y(x) \), integrate the right-hand side of the equation with respect to \( x \): \[ y = \int \left( \frac{2}{x^3} - x^3 \right) \, dx\] Split the integral into two separate integrals: \[ y = \int \frac{2}{x^3} \, dx - \int x^3 \, dx\]
3Step 3: Solve Each Integral Separately
Calculate the first integral: \[ \int \frac{2}{x^3} \, dx = 2 \int x^{-3} \, dx = 2 \left(-\frac{1}{2}x^{-2}\right) = -\frac{1}{x^2}\]Calculate the second integral using the power rule: \[ \int x^3 \, dx = \frac{x^4}{4} \]
4Step 4: Combine the Results
Substitute the results from the individual integrals into the expression for \( y \): \[ y = -\frac{1}{x^2} - \frac{x^4}{4} + C\] Here, \( C \) is the constant of integration.
5Step 5: Write Down the General Solution
The general solution to the differential equation is: \[ y(x) = -\frac{1}{x^2} - \frac{x^4}{4} + C\] where \( C \) is any constant.
Key Concepts
First-Order Differential EquationsIntegration TechniquesGeneral Solution
First-Order Differential Equations
A first-order differential equation involves the first derivative of a function, but no higher derivatives. In the given problem, the equation \( \frac{d y}{d x} = \frac{2}{x^3} - x^3 \) is a classic example because it only involves \( \frac{d y}{d x} \), the first derivative of \( y \) with respect to \( x \).
What makes first-order differential equations interesting is their wide applicability, ranging from simple physical processes like cooling of objects to more complex dynamics found in engineering and biological systems.
To solve such equations, we often aim to express the derivative in terms of \( x \) and integrate both sides to find the original function \( y \). The main goal is to eliminate the derivative, leaving us with a meaningful expression for \( y(x) \) that satisfies the original differential equation.
What makes first-order differential equations interesting is their wide applicability, ranging from simple physical processes like cooling of objects to more complex dynamics found in engineering and biological systems.
To solve such equations, we often aim to express the derivative in terms of \( x \) and integrate both sides to find the original function \( y \). The main goal is to eliminate the derivative, leaving us with a meaningful expression for \( y(x) \) that satisfies the original differential equation.
Integration Techniques
Integration is a fundamental tool in solving first-order differential equations. The given solution shows two integrals: \( \int \frac{2}{x^3} \, dx \) and \( \int x^3 \, dx \).
To tackle these integrals, we first split the equation into simpler parts. For instance:
To tackle these integrals, we first split the equation into simpler parts. For instance:
- \( \int \frac{2}{x^3} \, dx = 2 \int x^{-3} \, dx \) requires rewriting the fraction to a power of \( x \).
- The technique used here is called power rule integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \), valid for all \( n eq -1 \).
General Solution
Once all parts of a differential equation are integrated, they are combined to form the general solution. In our example, combining each integral gives us the expression \( y = -\frac{1}{x^2} - \frac{x^4}{4} + C \), where \( C \) is a constant.
The presence of \( C \) is crucial as it represents the family of curves, indicating multiple solutions depending on initial conditions. That's why it's called a 'general' solution. In real-world applications, specific conditions or additional constraints will help determine this constant, thereby narrowing down to a particular solution.
Understanding the general solution helps us envision a broad spectrum of possible functions that satisfy the differential equation. This versatility is why differential equations are such powerful tools in mathematical modeling.
The presence of \( C \) is crucial as it represents the family of curves, indicating multiple solutions depending on initial conditions. That's why it's called a 'general' solution. In real-world applications, specific conditions or additional constraints will help determine this constant, thereby narrowing down to a particular solution.
Understanding the general solution helps us envision a broad spectrum of possible functions that satisfy the differential equation. This versatility is why differential equations are such powerful tools in mathematical modeling.
Other exercises in this chapter
Problem 47
Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Assume further that \(f(a)=f(b)=0\) but \(f\) is not constant on \([a, b] .\) Ex
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Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(\frac{x}{1+x}\right)^{x} $$
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