Page 61 - Schaum's Outline of Differential Equations

44 LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 6

so (6.2) becomes

6.6. Solve the differential equation in the previous problem.
Multiplying the differential equation by the integrating factor defined by (_/) of Problem 6.5, we obtain

Integrating both sides of this last equation with respect to x, we obtain

6.7. Solve y + y = sin x.
ldx
Here p(x) = 1; hence I(x) = &> = g*. Multiplying the differential equation by I(x), we obtain

Integrating both sides of the last equation with respect to x (to integrate the right side, we use integration by parts
twice), we find

6.8. Solve the initial-value problem / + y = sin x; y(n) = 1.
From Problem 6.7, the solution to the differential equation is

Applying the initial condition directly, we obtain

Thus

6.9. Solve
5
(
dx
5x
Here p(x) = -5 and I(x) = e - ~> = g , Multiplying the differential equation by I(x), we obtain
Sx
5x
Integrating, we obtain ye = c or y = ce .
Note that the differential equation is also separable. (See Problem 4.4.)

6.10. Solve
This is a linear differential equation for the unknown function z(x). It has the form of Eq. (6.1) with y replaced
by z and p(x) = q(x) = —x. The integrating factor is