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 $y = \sqrt{- \cos^{2} x \sin^{2} x + C \sin^{4} x}$ .

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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

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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

Engineering Physics

Fluids

Torque and Rotational Motion

Translational Dynamics

Oscillations

Electricity

Study anywhere. Anytime. Across all devices.

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