Page 38 - Advanced engineering mathematics

18 CHAPTER 1 First-Order Differential Equations

Integrate to obtain
3
4
xy = x + c.
4
Solve for y to write the general solution
3 c
3
y = x +
4 x
for x > 0. For the initial condition, we need
3
y(1) = + c = 5.
4
Then c = 17/4, and the solution of the initial value problem is
3 17
3
y = x + .
4 4x
As suggested previously, solving a linear differential equation may lead to integrals we can-

not evaluate in elementary form. As an example, consider y + xy = 2. Here p(x) = x, and an
integrating factor is
2
xdx x /2
e = e .
Multiply the differential equation by the integrating factor:
2
2
x /2
y e 2 + xe x /2 y = 2e x /2 .
Write the left side as the derivative of a product:
d x /2
x /2
2
2
e y = 2e .
dx
Integrate

2
2
ye x /2 = 2 e x /2 dx + c.
The general solution is

2
2
2
y = 2e −x /2 e x /2 dx + ce −x /2 .
2
x /2
We cannot evaluate e dx in elementary terms (as a finite algebraic combination of elemen-
2
tary functions). We could do some additional computation. For example, if we write e x /2 as a
power series about 0, we could integrate this series term by term. This would yield an infinite
series expression for the solution.
Here is an application of linear equations to a mixing problem.
EXAMPLE 1.10 A Mixing Problem
We want to determine how much of a given substance is present in a container in which var-
ious substances are being added, mixed, and drained out. This is a mixing problem, and it is
encountered in the chemical industry, manufacturing processes, swimming pools and (on a more
sophisticated level) in ocean currents and atmospheric activity.
As a specific example, suppose a tank contains 200 gallons of brine (salt mixed with water)
in which 100 pounds of salt are dissolved. A mixture consisting of 1/8 pound of salt per gallon

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October 14, 2010 14:9 THM/NEIL Page-18 27410_01_ch01_p01-42