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

Given information

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

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.

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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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

Step by step solution

Given information

Definition of differential equation

Solve the differential equation

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Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.y'=yx-tanyx

Short Answer

Step by step solution

Given information

Definition of differential equation

Solve the differential equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Physical Quantities And Units

Further Mechanics and Thermal Physics

Famous Physicists

Turning Points in Physics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

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