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Chapter 5: Problem 1

Solve the differential equation \(2 t+3 x^{2} \dot{x}=0\), first as a separable equation, and second by considering it as an exact equation.

Short Answer

Expert verified
The separable solution of the differential equation is given by: \[ x^{3} + t^{2}=C \]. It can also be found as an exact equation.

Step by step solution

01

Recognize the Form

The given differential equation is: \[ 2 t + 3 x^{2} \frac{dx}{dt} = 0 \]This can be solved by the method of separation of variables or by treating it as an exact equation.
02

Rearrange as a Separable Equation

Rewrite the equation in a form that allows the separation of variables:\[ 3 x^{2} \frac{dx}{dt} = -2t \]Then, separate the variables, placing all terms involving \(x\) on one side and all terms involving \(t\) on the other side:\[ 3 x^{2} dx = -2t dt \]
03

Integrate Both Sides

Integrate both sides with respect to their respective variables:\[ \begin{aligned} \text{Left Side:} & ewline \text{ } ewline \text{ } ewline \text{ } ewline \text{ } ewline \text{ } ewline & ewline & abla &= \text{integrate } 3 x^{2} = \text{ Integrate } -2 t \] For detailed computation, it leads to \[ x^{3} =-\frac{t^{2} }{3} + C \]
04

Solve for the Constant of Integration

Integrate both sides:\[ \frac{3 x^{3}}{3} = -\frac{2 t^2}{2} + C \]ewline leading to \[ x^{3} = -\frac{t^2}{3} + C \]
05

Considering the DE as an Exact Equation

Rewrite the differential equation and verify the exactness:\[ 3x^2dx + 2t dt = 0 \] For verification \frac{\right}i.e., (\textbf M)}dx)dx+and findtex \end aligns and multiplies along its respected axes so to obey solutions of selection (\textbf N)}dt)dt + C to get margins resulting @ (\circ)\textbf{}

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Key Concepts

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

separable equations
A separable equation is a type of differential equation that can be written in a form where the variables can be separated on opposite sides of the equation. To solve the given differential equation using separation of variables, we started with:






Then, we rewrote it to:


The goal was to get all expressions involving x on one side and all expressions involving t on the other side. This yielded the new form:

  • \[ 3 x^{2} dx = -2t dt \]


We now integrate both sides with respect to their respective variables:

  • The integral of \( 3 x^{2} \) with respect to \( x \) yields:
    \[ \frac{3}{3} x^{3} = x^{3} \]
  • The integral of \( -2t \) with respect to \( t \) yields:
    \[ - \frac{2}{2} t^{2} = -t^{2} \]

Which gives:

  • \[ x^{3} = -\frac{t^{2}}{3} + C \]
.
Here \( C \) is the constant of integration.
exact equations
An exact equation is another way to solve certain differential equations. A differential equation of the form

\[ \frac{dx}{dt} + g(x,t) = 0 \]
is considered exact if there exists a function \( F(x,t) \) such that: \( F_x = M \)
and \( F_t = N \).

For our example, we look at the original equation: \[ 3x^2 dx + 2t dt = 0 \]. We can verify the exactness by checking if there exists some function \( F(x,t) \) where:
  • \( F_x = 3x^2 \)
  • and \( F_t = 2t \)
If these conditions are met, \( F \) is the potential function, and thus the differential equation is exact. This verifies the consistency with both sides of our initial integration.
integration constants
When solving differential equations, we often end up integrating one or more times. Each integration introduces a constant 鈥� called the constant of integration 鈥� because integration is the reverse operation of differentiation, which means there is always some arbitrary constant added during integration. Take, for example, our separated equation:

\( \frac{3 x^3}{3} = -\frac{2 t^2}{2} + C \)
Here, the \( C \) is our constant of integration.
This constant represents an infinite number of possible vertical shifts of our solution curve. To find a specific solution, we often need an initial condition that gives us a specific value for C.
integration techniques
Using different integration techniques can help solve differential equations more efficiently. In our example, straightforward indefinite integration was used. We take separate integrals:
  • For the left side, we integrated \( 3 x^{2} \), resulting in \( \frac{3}{3} x^{3} = x^{3} \)
  • For the right side, integrating \( -2t \) yields: \( - \frac{2}{2} t^{2} = -t^{2} \)
These common techniques include:

  • Substitution: Changing variables to simplify the integration.
  • Partial Fraction Decomposition: Used when integrating rational functions.
  • Integration by Parts: Often used when the integrand is a product of functions.
Learning these methods can make solving differential equations less challenging. They provide flexibility and quicker solutions for various types of differential equations.

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Most popular questions from this chapter

Find the general solution of \(\dot{x}+\frac{1}{2} x=\frac{1}{4}\). Determine the equilibrium state of the equation, and examine whether it is stable. Also draw some typical integral curves.

Solve the differential equation \(1+(2+t / x) \dot{x}=0, t>0, x>0\)

An economic growth model by T. Haavelmo (1954) leads to the differential equation $$ \dot{K}=\gamma_{1} b K^{\alpha}+\gamma_{2} K $$ where \(\gamma_{1}, \gamma_{2}, b\) and \(\alpha\) are positive constants, \(\alpha \neq 1\) and \(K=K(t)\) is the unknown function. The equation is separable, but solve it as a Bernoulli equation.

Show that \(x=C t^{2}\) is a solution of the differential equation \(t \dot{x}=2 x\) for all choices of the constant \(C\). Find in particular the integral curve through \((1,2)\).

Show that any function \(x=x(t)\) that satisfies the equation \(x e^{t x}=C\) is a solution of the differential equation \((1+t x) \dot{x}=-x^{2} .\) (Hint: Differentiate \(x e^{t x}=C\) implicitly w.r.t. \(t .\).)
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