/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Solve the differential equation ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

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

Short Answer

Expert verified
Rewritten as \( \frac{dx}{dt} = - \frac{x}{2+t/x} \), separate and integrate: \( \frac{dx}{x} = - \frac{dt}{2+t/x} \).

Step by step solution

01

- Rewrite the given differential equation

Start by rewriting the differential equation in a more recognizable form for solving.
02

- Separate variables

Isolate the terms involving \(x\) and \(t\). This often involves moving terms around to get one side of the equation to be dependent on \(x\) and the other on \(t\).
03

- Integrate both sides

Integrate both sides with respect to their respective variables.
04

- Solve for x

Solve the resulting equation for \(x\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Separating Variables
In solving differential equations, one common technique is called 'separating variables.' This method is particularly useful when you can rearrange the equation so that all terms involving one variable (say, x) are on one side of the equation, and all terms involving another variable (say, t) are on the other side. Here's a step-by-step explanation for separating variables:
  • Rewrite the equation if necessary. This might involve simplifying or expanding it.
  • Move all terms involving x to one side of the equation and all terms involving t to the other side. For example, in the differential equation given
    鈧佲倞鈧嶁倐鈧娾倻/鈧撯値 鈧撎�=0, you first rewrite it as:
    鈧撎� = -鈧嶁倐鈧娾倻/鈧撯値.
Once variables are separated, you can proceed to the next step, which is integration. This method simplifies complex differential equations by breaking them down into more manageable parts.
Integration
After you have separated variables, the next crucial step is integration. Integrating both sides with respect to their respective variables involves the following steps:
  • For the side involving x, integrate with respect to x.
  • For the side involving t, integrate with respect to t.
  • Remember to include the constant of integration (C) after performing the integration.
Following this process, we have the equation:
鈭倣1/鈧撯値dx = -鈭倣鈧傗倞鈧溾値dt. Integration simplifies our separated equation into something much more straightforward, ultimately leading us to an expression that we can solve for x.
Solving Differential Equations
The last step in solving a differential equation is to solve for the unknown function, in this case, x. After integrating both sides, we typically obtain an equation involving x and the variable t. Here's how you proceed:
  • Combine like terms, simplify and isolate x on one side of the equation.
  • Solve the equation algebraically to find x as a function of t.
  • In some cases, you might need to apply an initial condition to find the particular solution.
For our example, the integration step might yield something like:
ln|x| = -2t + C. To solve for x, we would exponentiate both sides:
e^(ln|x|) = e^(-2t + C), yielding:
x = e^C e^(-2t). Hence, you obtain:
x(t) = Ce^(-2t), where C is a constant determined by initial conditions. Following these steps ensures a structured approach to solving differential equations effectively and accurately.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the following. Bernoulli equations assuming \(t>0, x>0\) : (a) \(t \dot{x}+2 x=t x^{2}\) (b) \(\dot{x}=4 x+2 e^{t} \sqrt{x}\) (c) \(t \dot{x}+x=x^{2} \ln t\)

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.

Find the general solution of \(\dot{x}+a(t) x=0\). In particular, when \(a(t)=a+b c^{t}(a, b\), and \(c\) are positive; \(c \neq 1\) ) show that the solution of the equation can be written in the form \(x=C p^{t} q^{c^{t}}\), where \(p\) and \(q\) are constants determined by \(a, b\), and \(c\), whereas \(C\) is an arbitrary constant. (This is Gompertz-Makeham's law of mortality.)

Solve the differential equation \((1+t x) \dot{x}=-x^{2}\). (Hint: Try the substitution \(w=t x\).)

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. \(t .\).)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.