Chapter 8: Q13P (page 406)

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.
$y y' - 2 y^{2} c o t x = s i n x c o s x$

Short Answer

Expert verified

Answer

The general solution of the differential equation is $y = \sqrt{- \cos^{2} x c i n^{2} x + C \sin^{4} x}$

Step by step solution

Given information

The given differential equation is $y y' - 2 y^{2} c o t x = \sin x \cos x$ .

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

Rewrite the differential equation as,

$y' - 2 y c o t x = \frac{\sin 2 x}{2 y}$

This equation is in the form of Bernoulli equation. So we can solve it by assuming $z = y^{2}$ . Then, $z' = 2 y y'$ .

$2 y y' - 4 y^{2} c o t^{2} c o t x = \sin 2 x z' - 4 z c o t x = \sin 2 x$

Now it becomes a first order linear differential equation as,

$I = - 4 \int c o t x d x = - 4 I n (\sin x) e^{I} = \sin^{- 4} x$

Solve the equation further as,

$z e^{I} = (\sin 2 x) \sin^{4} x d x = - c o t^{2} x + C z = - \cos^{2} x \sin^{2} x + C \sin^{4} x$

Finally, substitute $z = y^{2}$ . So, the solution is $y = \sqrt{- \cos^{2} x \sin^{2} x + C \sin^{4} x}$ .

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.
$y y' - 2 y^{2} c o t x = s i n x c o s x$

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

Thermodynamics

Conservation of Energy and Momentum

Scientific Method Physics

Physics of Motion

Fluids

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.yy'-2y2cotx=sinxcosx

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

Thermodynamics

Conservation of Energy and Momentum

Scientific Method Physics

Physics of Motion

Fluids

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.
$y y' - 2 y^{2} c o t x = s i n x c o s x$