Page 214 - Schaum's Outline of Differential Equations
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CHAP. 20] SOLVING SECOND-ORDER DIFFERENTIAL EQUATIONS 197
Solved Problems
20.1. Reduce the initial-value problem y" -y = x; y(0) = 0, / (0) = 1 to System (20.7).
Defining z = y', we have z(0) = y'(0) = 1 and z' = y". The given differential equation can be rewritten as
y" = y + x, or z' = y + x. We thus obtain the first-order system
20.2. Reduce the initial-value problem /' - 3/ + 2y = 0; y(0) = -I, y'(0) = 0 to System (20.7).
Defining z = y', we have z(0) = y'(0) = 0 and z' = y". The given differential equation can be rewritten as
y" = 3y' — 2y, or z' = 3z — 2y. We thus obtain the first-order system
20.3. Reduce the initial-value problem 3x y" - xy' + y = 0; y(l) = 4, /(I) = 2 to System (20.7).
2
Defining z = y', we have z(l) = y'(l) = 2, and z' = y". The given differential equation can be rewritten as
or
We thus obtain the first-order system
2
20.4. Reduce the initial-value problem y'" - 2xy" + 4y'-x y=l; y(0) = 1, /(O) = 2, /'(O) = 3 to System (20.2).
Following Steps 1 through 3 of Chapter 17, we obtain the system
To eliminate subscripting, we define y = y\,Z = y^, and w = y 3. The system then becomes
