Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.
$\frac{d x}{d t} = x^{2} - 2 y^{- 3}, \frac{d y}{d t} = 3 x^{2} - 2 x y$

The solution is $x^{3} - x^{2} y - y^{- 2} = c$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find phase plane equation

Here the system is;

$\frac{d x}{d t} = x^{2} - 2 y^{- 3} \frac{d y}{d t} = 3 x^{2} - 2 x y$

And the phase plane equation is;

$\frac{d y}{d x} = \frac{3 x^{2} - 2 x y}{x^{2} - 2 y^{- 3}}$

Step 2: Solve for exactness

Here the equation is $\frac{d y}{d x} = \frac{3 x^{2} - 2 x y}{x^{2} - 2 y^{- 3}}$ .

$\begin{array}{rcl} (2 x y - 3 x^{2}) d x + (x^{2} - 2 y^{- 3}) d y & = & 0 \\ M & = & (2 x y - 3 x^{2}) \\ N & = & (x^{2} - 2 y^{- 3}) \\ \frac{\partial M}{\partial y} & = & 2 x = \frac{\partial N}{\partial x} \end{array}$

Step 3: Find the value of F and G.

Now,

$\begin{array}{rcl} F (x, y) & = & \int M (x, y) d x + g (y) \\ = & \int (2 x y - 3 x^{2}) d x + g (y) \\ = & x^{2} y - x^{3} + g (y) \\ N (x, y) & = & x^{2} + g' (y) \\ x^{2} - 2 y^{- 3} & = & x^{2} + g' (y) \\ g' (y) & = & - 2 y^{- 3} \\ g (y) & = & y^{- 2} + c \\ F (x, y) & = & x^{3} - x^{2} y - y^{- 2} + c \end{array}$

Therefore, the solution is $x^{3} - x^{2} y - y^{- 2} = c$ .

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.
$\frac{d x}{d t} = x^{2} - 2 y^{- 3}, \frac{d y}{d t} = 3 x^{2} - 2 x y$

Step by Step Solution

Step 1: Find phase plane equation

Step 2: Solve for exactness

Step 3: Find the value of F and G.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.dxdt=x2-2y-3,dydt=3x2-2xy

Step by Step Solution

Step 1: Find phase plane equation

Step 2: Solve for exactness

Step 3: Find the value of F and G.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.
$\frac{d x}{d t} = x^{2} - 2 y^{- 3}, \frac{d y}{d t} = 3 x^{2} - 2 x y$