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Chapter 8: Problem 68

Solve the initial value problems in Exercises \(67-70\) for \(x\) as a function of \(t .\) $$ \left(3 t^{4}+4 t^{2}+1\right) \frac{d x}{d t}=2 \sqrt{3}, \quad x(1)=-\pi \sqrt{3} / 4 $$

Short Answer

Expert verified
Separate variables, integrate both sides, and apply initial conditions.

Step by step solution

01

Identify the Differential Equation

The given differential equation is \( (3t^4 + 4t^2 + 1) \frac{dx}{dt} = 2\sqrt{3} \). This is a first order linear differential equation.
02

Separate Variables

Separate the variables so that one goes to each side of the equation. Rewrite it as \( \frac{dx}{dt} = \frac{2\sqrt{3}}{3t^4 + 4t^2 + 1} \).
03

Integrate Both Sides

Integrate both sides to find \( x(t) \). Compute the integral \( \int dx = \int \frac{2\sqrt{3}}{3t^4 + 4t^2 + 1} \, dt \).
04

Simplify the Denominator

Factor the denominator if possible. In this case, it doesn't factor easily, so proceed by direct integration.
05

Use a Trigonometric Substitution

Since this integral appears complex, consider transformations like trigonometric substitution or refer to integral tables for complex polynomial denominators to solve for \( x(t) \).
06

Apply the Initial Condition

Use the initial condition \( x(1) = -\frac{\pi \sqrt{3}}{4} \) to determine the constant of integration. Substitute \( t = 1 \) and \( x = -\frac{\pi \sqrt{3}}{4} \) into the integral result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Order Linear Differential Equations
When dealing with initial value problems, it's crucial to recognize the type of differential equation at hand. In this context, we are given a first order linear differential equation. This type of equation is characterized by the presence of the derivative of the unknown function, typically denoted as \( \frac{dx}{dt} \), and a function of the independent variable, here being \( t \). For instance:
  • The format \( a(t) \frac{dx}{dt} + b(t)x = c(t) \) is typical.
  • In our exercise, it simplifies to \( \frac{dx}{dt} = \frac{2\sqrt{3}}{3t^4 + 4t^2 + 1} \).
Spotting this format is the first step in solving the equation, as it guides us toward using appropriate methods like separation of variables.
Trigonometric Substitution
Sometimes integrals involving polynomials can be challenging. This is where trigonometric substitution enters the scene. This method is used to simplify integrals by substituting trigonometric functions for algebraic expressions. For instance:
  • In cases where a denominator resembles a trigonometric identity, like \( a^2 + x^2 \), substitutions such as \( x = a \tan(\theta) \) may be made.
  • While our integral does not directly simplify through elementary substitutions, considering trigonometrically related forms or consulting integration tables can help streamline complex integrations.
By using trigonometric substitution, we can often transform difficult integrals to more manageable forms.
Integration Techniques
Solving the equation involves integrating the separated sides, which requires choosing the suitable method for integration. In our case, integrating \( \int \frac{2\sqrt{3}}{3t^4 + 4t^2 + 1} \, dt \) demands specific techniques:
  • You might need partial fraction decomposition if your polynomial factors nicely, simplifying the integral into a sum of simpler fractions.
  • Direct integration may be necessary if factorization is not feasible, along with exploring sophisticated methods like trigonometric substitutions.
Applying these techniques helps in computing integrals that don't immediately lend themselves to basic techniques, ensuring that we find the antiderivative essential for problem-solving.
Separation of Variables
One of the most straightforward methods for solving first order differential equations is separation of variables. The idea is to isolate the variables so that each side of the equation depends on only one variable. In our particular problem, this method works as follows:
  • Begin by rewriting the equation \( \frac{dx}{dt} = \frac{2\sqrt{3}}{3t^4 + 4t^2 + 1} \) to split \( dx \) and \( dt \).
  • Integrate both sides: \( \int dx = \int \frac{2\sqrt{3}}{3t^4 + 4t^2 + 1} \, dt \).
This separation allows for straightforward integration of both sides, eventually leading to a general solution. Applying the initial condition helps us find the particular solution that satisfies the problem's requirements.

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