Chapter 5: Q8E (page 271)

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

Short Answer

Expert verified

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

Step by step solution

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;

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

Solve for exactness

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

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

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

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

Short Answer

Step by step solution

Find phase plane equation

Solve for exactness

Find the value of F and G.

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Most popular questions from this chapter

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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

Short Answer

Step by step solution

Find phase plane equation

Solve for exactness

Find the value of F and G.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Decision Maths

Calculus

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

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