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PARTIAL DIFFERENTIAL EQUATION SOLUTION FIRST ORDER LINEAR
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ANALYTIC METHODS FOR EXACT AND LINEAR DIFFERENTIAL EQUATIONS

EXACT DIFFERENTIAL EQUATIONS

INTEGRATING FACTOR TO FORCE EXACTNESS

LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS

A first order linear equation has the following form.

(20) [Graphics:2txgr137.gif]

Note that [Graphics:2txgr138.gif] is a function of x alone. If [Graphics:2txgr139.gif], then the equation is homogeneous. If we rearange (20) as follows

(21) [Graphics:2txgr140.gif]

Comparing this form to (1) shows that the first order linear is very similar to the exact type differential equation. From this we see that

(22) [Graphics:2txgr141.gif]

and

(23) [Graphics:2txgr142.gif]

Applying the partial derivative test to this gives [Graphics:2txgr143.gif] and [Graphics:2txgr144.gif], as verified below. Unless [Graphics:2txgr145.gif] the linear first order will not pass the exactness test. This suggests we try the integrating factor method suggested above. Equation (18) is what we need. It is reproduced below.

	[Graphics:2txgr146.gif]
	[Graphics:2txgr147.gif]

Now looking at equation (18),

(24) [Graphics:2txgr148.gif]

we see that [Graphics:2txgr149.gif], [Graphics:2txgr150.gif], and [Graphics:2txgr151.gif]. This gives the following result for the integrating factor for the linear first-order equation.

(25) [Graphics:2txgr152.gif]

Multiplying (21) by this results gives the following:

(26) [Graphics:2txgr153.gif]

Notice that the left side is [Graphics:2txgr154.gif] so if we rewrite (26) and then integrate with respect to x we get

(27) [Graphics:2txgr155.gif].

The left side of this result is integrated by inspection to give

(28) [Graphics:2txgr156.gif]

or solving for y we have the following:

(29) [Graphics:2txgr157.gif]

Once a differential equation has been analyzed as first-order linear, then all that must be done is to identify the [Graphics:2txgr158.gif] and the [Graphics:2txgr159.gif] then substitute into (29) and do the integration. This result also constitutes a proof that a solution exists for the first order linear differential equation.Let us look at an example.

Example 3: [Graphics:2txgr160.gif]

In this case we also give an initial condition, that is [Graphics:2txgr161.gif]. First we rearange the equation to the form recognizable as first-order linear.

[Graphics:2txgr162.gif]

From this we see that [Graphics:2txgr163.gif] and [Graphics:2txgr164.gif]. We will try (29) on this problem. Because of similar names for several of the variables, I will turn off Mathematica's spell checker.

	[Graphics:2txgr165.gif]

	[Graphics:2txgr166.gif]
	[Graphics:2txgr167.gif]

Next we must satisfy the initial condition in order to evaluate the constant of integration, CC3.

	[Graphics:2txgr168.gif]
	[Graphics:2txgr169.gif]

	[Graphics:2txgr170.gif]
	[Graphics:2txgr171.gif]

We will plot our solution near the initial condition point (1,1).

	[Graphics:2gr173.gif]

	[Graphics:2gr174.gif]

                [Graphics:2gr175.gif]

For comparison, we will look at the DSolve results, which is the same as above.

	[Graphics:2txgr175.gif]
	[Graphics:2txgr176.gif]

	[Graphics:2txgr177.gif]

EXERCISES: Solve and plot when initial conditions are given.

1 [Graphics:2txgr178.gif] 2 [Graphics:2txgr179.gif] 3 [Graphics:2txgr180.gif] 4 [Graphics:2txgr181.gif]

SOLUTIONS

1 [Graphics:2txgr182.gif]

	[Graphics:2txgr183.gif]

	[Graphics:2txgr184.gif]
	[Graphics:2txgr185.gif]

	[Graphics:2txgr186.gif]
	[Graphics:2txgr187.gif]

	[Graphics:2txgr188.gif]

                [Graphics:2txgr189.gif] 2` [Graphics:2txgr190.gif]

	[Graphics:2txgr191.gif]

	[Graphics:2txgr192.gif]
	[Graphics:2txgr193.gif]
3 [Graphics:2txgr194.gif]

	[Graphics:2txgr195.gif]

	[Graphics:2txgr196.gif]
	[Graphics:2txgr197.gif]

	[Graphics:2txgr198.gif]
	[Graphics:2txgr199.gif]

	[Graphics:2txgr200.gif]

                [Graphics:2txgr201.gif] 4 [Graphics:2txgr202.gif]

	[Graphics:2txgr203.gif]

	[Graphics:2txgr204.gif]
	[Graphics:2txgr205.gif]

	[Graphics:2txgr206.gif]
	[Graphics:2txgr207.gif]

	[Graphics:2txgr208.gif]

                [Graphics:2txgr209.gif]

SPECIAL EXAMPLE I: Products vrs the Environment

SPECIAL EXAMPLE II: Nuclear Chemistry