Page 63 - Schaum's Outline of Differential Equations

46 LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 6

Multiplying the differential equation by I(t), we obtain

or

Upon integrating both sides of this last equation, we have

whereupon

6.14. Solve the initial-value problem

The solution to this differential equation is given in Problem 6.13 as

Applying the initial condition directly, we have

6.15. Solve

This is a linear differential equation for the unknown function T(t). It has the form of Eq. (6.1) with y replaced
by T, x replaced by t, p(t) = k, and q(t) = 100k. The integrating factor is

Multiplying the differential equation by I(t), we obtain

or

Upon integrating both sides of this last equation, we have

whereupon

6.16 Solve y + xy = xy2
This equation is not linear. It is, however, a Bernoulli differential equation having the form of Eq. (6.4) with
1 2
l
p(x) = q(x) = x, and n = 2. We make the substitution suggested by (6.5), namely, z = y ^ = y~ , from which follow