A Solution of Euler's Type for an Exact Differential Equation

choose function f(x)

1

2

3

tangent length

0.3

contours

1

step

11

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

x

y

An equation of the form

f(x,y)=C

is the general solution of the exact differential equation

f

x

dx+f

y

dy=0

(or

gradf·(dx,dy)=0

). So the contours of the Mathematica built-in function ContourPlot are particular solutions of the equation. To find a graphical solution (of Euler type) of the equation, choose an initial point using the locator, the length of the tangent, and then proceed along the green arrow and add new locators. The green arrow is perpendicular to the red arrow that shows the direction of the gradient. Check the solution by increasing the number of contours. To construct Euler's polygonal line, choose a sufficiently long tangent line and the step given by vertical mesh lines.

The choices for the function

f(x,y)

are:

(1)

x

2

+0.9y

2

-2x+y

,

(2)

x

2

+2y

,

(3)

x

2

+2y

2

-x

.