Page 92 - Schaum's Outline of Differential Equations

CHAP. 8] LINEAR DIFFERENTIAL EQUATIONS: THEORY OF SOLUTIONS 75

Solved Problems

8.1. State the order of each of the following differential equations and determine whether any are linear:

2
(a) 2xy" + x y' - (sin x)y = 2 (b) yy'" + xy'+ y = x 2
(c) y"-y=0 (d) 3y'+xy = e~* 2

x
x
(e) 2e y'" + e y" = 1
(h)

2
(a) Second-order. Here b 2(x) = 2x, b^x) = x , b Q(x) = -sin x, and g(x) = 2. Since none of these terms depends on
y or any derivative of y, the differential equation is linear.
(b) Third-order. Since b 3 = y, which does depend on y, the differential equation is nonlinear.
(c) Second-order. Here b 2(x) = 1, bi(x) = 0, b 0(x) = 1, and g(x) = 0. None of these terms depends on y or any
derivative of y; hence the differential equation is linear.
x
(d) First-order. Here b 1(x) = 3, b Q(x)=x, and g(x) = e~ ; hence the differential equation is linear. (See also
Chapter 5.)
x
x
(e) Third-order. Here b 3(x) = 2e , b 2(x) = e , bi(x) = b Q(x) = 0, and g(x) = 1. None of these terms depends on y or
any of its derivatives, so the equation is linear.
(/) Fourth-order. The equation is nonlinear because y is raised to a power higher than unity.
(g) Second-order. The equation is nonlinear because the first derivative of y is raised to a power other than unity,
here the one-half power.
(h) First-order. Here bi(x) = 1, b 0(x) = 2, and g(x) = -3. None of these terms depends on y or any of its derivatives,
so the equation is linear.

8.2. Which of the linear differential equations given in Problem 8.1 are homogeneous?
Using the results of Problem 8.1, we see that the only linear differential equation having g(x) = 0 is (c), so
this is the only one that is homogeneous. Equations (a), (d), (e), and (h) are nonhomogeneous linear differential
equations.

8.3. Which of the linear differential equations given in Problem 8.1 have constant coefficients?
In their present forms, only (c) and (h) have constant coefficients, for only in these equations are all the
coefficients constants. Equation (e) can be transformed into one having constant coefficients by multiplying it by
x
e~ . The equation then becomes

8.4. Find the general form of a linear differential equation of (a) order two and (b) order one.
(a) For a second-order differential equation, (8.1) becomes

If b 2(x) 2 0, we can divide through by it, in which case (8.3) takes the form