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Chapter 8: Q12P (page 406)

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.
$y' = \frac{y}{x} - t a n \frac{y}{x}$

Short Answer

Expert verified

Answer

The general solution of the differential equation is $y = x \sin^{- 1} \frac{C_{1}}{x}$

Step by step solution

01

Given information

The given differential equation is $y' = \frac{y}{x} - \tan \frac{y}{x}$ .

02

Definition of differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.

03

Solve the differential equation

Since $y' = f \frac{y}{x}$ , therefore we can solve this differential equation as a homogeneous equation. Assume $v = \frac{y}{x}$ . Then, $y' = x v' + v$ . Therefore,

$x v' + v = v - \tan v v' = - \frac{\tan v}{x}$

, which has become a separable equation and its solution of is,

$\frac{d v}{\tan v} = - \frac{d x}{x} I n (S i n v) = - I n x + C \sin v = \frac{C_{1}}{x}$

Substitute $v = \frac{y}{x}$ . Therefore, the solution becomes,

$y = x \sin^{- 1} \frac{C_{1}}{x}$

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Most popular questions from this chapter

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y " + 16 y = 8 c o s 4 t, y_{0} = 0, 蠄'_{0} = 8$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y " + 16 y = 8 c o s 4, y_{0} = y_{0}^{'} = 0$

Use L28 to find the Laplace transform of

$f (t) = {\begin{cases} s i n (t - 蟺 / 2), & t > 蟺 / 2 . \\ 0, & t < 蟺 / 2 . \end{cases}$

Heat is escaping at a constant rate [ $\frac{d Q}{d t}$ in $(1.1)$ is constant] through the walls of a long cylindrical pipe. Find the temperature T at a distance r from the axis of the cylinder if the inside wall has radius $r = 1$ and temperature $T = 100$ and the outside wall has $r = 2$ and $T = 0$

Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.

$(x l n x) y + y = I n x$

See all solutions

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