Chapter 5: Q9E (page 271)

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

Short Answer

Expert verified

The solution is $e^{x} - x y - y^{2} + c$ .

Step by step solution

Find phase plane equation

Here the system is;

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

And the phase plane equation is;

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

Solve the equation

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

role="math" localid="1663962144511" $\begin{array}{rcl} (e^{x} + y) d x + (x - 2 y) d y & = & 0 \\ M & = & (e^{x} + y) \\ N & = & (x - 2 y) \\ \frac{\partial M}{\partial y} & = & 1 = \frac{\partial N}{\partial x} \end{array}$

The equations are exact.

Find the value of F and G.

Now,

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

Therefore, the solution is $e^{x} - x y - y^{2} + c$ .

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Find phase plane equation

Solve the equation

Find the value of F and G.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Probability and Statistics

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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

Short Answer

Step by step solution

Find phase plane equation

Solve the equation

Find the value of F and G.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Probability and Statistics

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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