244 Ordinary Differential Equations
SOLUTION
Apply the Laplace Transform to the Differential Equation
Taking the Laplace transform of given differential equation, we find.C[y" + 4y] = .C[O].
Using the fact that .C[O] = 0 and the linearity property of the Laplace trans-
form, yields
.C[y"(x)] + 4.C[y(x)] = 0.
Replacing .C[y"(x)] by the expression we obtain from equation (2) with n = 2,
results in
-y'(O) - sy(O) + s^2 .C[y(x) ] + 4.C[y(x)] = 0
or
(3) -y'(O) - sy(O) + (s^2 + 4).C[y(x)] = 0.
Solve the Algebraic Equation for .C[y(x)]
Since specific initial conditions are not given, we let A = y(O) and B = y' (0),
where A and B are arbitrary real constants. Then solving equation (3) for
.C[y(x)], we obtain
(4)Apply the Inverse Laplace Transform
Applying the inverse Laplace transform to ( 4) and using the linearity of the
inverse Laplace transform, results in
y(x) = .c-^1 [B +As] = B.C_^1 [-1-] + A.C_^1 [-s-]
s^2 + 4 s^2 + 4 s^2 + 4= ~ sin2x + Acos2x.
Hence, the general solution of the differential equation y" + 4y = 0 is
y(x) = Csin2x +A cos 2x,
where A and C = B /2 are arbitrary real constants.
The following example shows how to use the Laplace transform method to
obtain the general solution of a second-order linear nonhomogeneous differen-
tial equations with constant coefficients.