Q33E In problems 33-40, Solve the equ... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 2: Q33E (page 76)

In problems 33-40, Solve the equation given in Problem 1.
$2 t x 鈥� d x + (t^{2} - x^{2}) d t = 0$

Short Answer

Expert verified

$x^{2} + t^{2} - C t = 0$ and $t 鈮� 0$

Step by step solution

Solving the given equation

The given equation is,

$2 t x dx + (t^{2} - x^{2}) dt = 0$

Simplify the above equation,

$\begin{array}{l} \frac{dx}{dt} = \frac{- (t^{2} - x^{2})}{2 tx} \\ \frac{dx}{dt} = \frac{x}{2 t} - \frac{t}{2 x} \end{array}$

Let,

localid="1663928794004" $\begin{array}{l} v = \frac{x}{t} \\ x = vt \\ \frac{dx}{dt} = v + t \frac{dv}{dt} \end{array}$

Substitute $\frac{x}{t} = v$ the $\frac{dx}{dt} = v + t \frac{dv}{dt}$ and in the equation (1),

$v + t \frac{dv}{dt} = \frac{v}{2} - \frac{1}{2 v}$

Simplify the above equation,

$\begin{array}{l} t \frac{dv}{dt} = \frac{v}{2} - \frac{1}{2 v} - v \\ t \frac{dv}{dt} = \frac{v^{2} - 1 - 2 v^{2}}{2 v} \\ t \frac{dv}{dt} = \frac{- 1 - v^{2}}{2 v} \\ t \frac{dv}{dt} = - \frac{1 + v^{2}}{2 v} \end{array}$

Cross multiplication on both sides in the above equation,

$\begin{matrix} t \frac{dv}{dt} = - \frac{1 + v^{2}}{2 v} \\ \frac{2 v}{1 + v^{2}} dv = - \frac{dt}{t} \end{matrix}$

Integrating

Taking integration both sides in the above equation

$鈭� \frac{2 v}{1 + v^{2}} dv = - 鈭� \frac{dt}{t}$

Solve the above equation,

$\begin{array}{l} \ln (1 + v^{2}) = - \ln (t) + \ln (C) \\ \ln (1 + v^{2}) + \ln (t) = \ln (C) \\ (1 + v^{2}) t = C \end{array}$

Finding the solution

Substitute the $v = \frac{x}{t}$ in the above equation,

$[1 + {(\frac{x}{t})}^{2}] t = C$

Simplify the above equation,

$\begin{array}{l} \frac{(t^{2} + x^{2}) t}{t^{2}} = C \\ x^{2} + t^{2} = C t \\ x^{2} + t^{2} - C t = 0 \end{array}$

Also note that is t=0 a solution

Hence the solution $x^{2} + t^{2} - C t = 0$ is and $t 鈮� 0$

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91影视

In problems 33-40, Solve the equation given in Problem 1.
$2 t x 鈥� d x + (t^{2} - x^{2}) d t = 0$

Short Answer

Step by step solution

Solving the given equation

Integrating

Finding the solution

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91影视

In problems 33-40, Solve the equation given in Problem 1.2tx鈥�dx+(t2-x2)dt=0

Short Answer

Step by step solution

Solving the given equation

Integrating

Finding the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Logic and Functions

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

In problems 33-40, Solve the equation given in Problem 1.
$2 t x 鈥� d x + (t^{2} - x^{2}) d t = 0$