Preliminaries. We wish to solve the exact D. E. with a given I. C.
,
it is necessary that .

The general solution is a family of curves f[x,y] = c, and the
particular solution is an implicit curve ,
where .

Computer Lab Work.

Load the following Mathematica graphics package.

.
 

Exercise 1. Use Mathematica to solve the exact D. E.
.
First, enter the functions m[x,y] and n[x,y].

Second, find the partial derivatives and verify that the D. E. is exact,
i.e. check to see that .

Third, integrate m[x,y] with respect to x to form the function f[x,y]
be sure to add the constant of integration g[y] that could involve y.

Fourth, to determine g[y] we need to solve the equation
.

Fifth, integrate g'[y] to form g[y].

Note that Mathematica has made the substitution in our formula for f[x,y].

Now form the general solution to the exact D. E.

Plot several curves in the family of solutions with Mathematica's generic command.

Plot several curves and specify the range of constants to be used

Plot the particular contour that corresponds to the constant c = f[1,1] = 4.

Exercise 2. Use Mathematica to solve the exact D. E.
.
First, enter the functions m[x,y] and n[x,y].

Second, find the partial derivatives and verify that the D. E. is exact,
i.e. check to see that .

Third, integrate m[x,y] with respect to x to form the function f[x,y]
be sure to add the constant of integration g[y] that could involve y.

Fourth, to determine g[y] we need to solve the equation
.

Fifth, integrate g'[y] to form g[y].

Note that Mathematica has made the substitution in our formula for f[x,y].

Now form the general solution to the exact D. E.

Plot several curves in the family of solutions with Mathematica's generic command.

Plot several curves and specify the range of constants to be used

Plot the particular contour that corresponds to the constant c = f[0,0] = 0.

Solutions.

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(c) John H. Mathews, 1998