2. Write the ODE as an equation of differentials. It should be exact. Find an integral F. Plot at least 5contours of F. Edit the plot, inserting arrows (like I did in class), place them on the contoursshowing the direction of flow.dyЗх-4ydx4х-5у

Answered: Write the ODE as an equation of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Write the ODE as an equation of differentials. It should be exact. Find an integral F. Plot at least 5
contours of F. Edit the plot, inserting arrows (like I did in class), place them on the contours
showing the direction of flow.
dy
Зх-4y
dx
4х-5у

Expert Solution

Step 1

Given differential equation is

$\frac{d y}{d x} = \frac{3 x - 4 y}{4 x - 5 y}$

equation of differentials

$y' (4 x - 5 y) = 3 x - 4 y$

now solution

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{3 x - 4 y}{4 x - 5 y} \\ y & = & v x, \frac{d y}{d x} = v + x \frac{d v}{d x} \\ v + x \frac{d v}{d x} & = & \frac{3 x - 4 v x}{4 x - 5 v x} = \frac{3 - 4 v}{4 - 5 v} \\ x \frac{d v}{d x} & = & \frac{3 - 4 v}{4 - 5 v} - v = \frac{3 - 4 v - 4 v + 5 v^{2}}{4 - 5 v} \\ \int \frac{- 2 (4 - 5 v)}{5 v^{2} - 8 v + 3} d v & = & \int \frac{- 2 d x}{x} + \ln k \end{array}$

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