To find the general solution of the second-order differential equation given by:
y” + 12y = 0
we begin by identifying the form of the equation. This is a linear homogeneous differential equation with constant coefficients. The standard approach is to assume a solution of the form:
y(t) = e^{rt}
where r is a constant to be determined. By substituting this assumed solution into the differential equation, we need to find the characteristic equation. First, we will differentiate:
y'(t) = r e^{rt}
y”(t) = r^2 e^{rt}
Substituting y(t), y'(t), and y”(t) into the differential equation:
r^2 e^{rt} + 12 e^{rt} = 0
Factoring out e^{rt} which is never zero gives us:
e^{rt}(r^2 + 12) = 0
This leads us to the characteristic equation:
r^2 + 12 = 0
To solve for r, we rewrite this equation:
r^2 = -12
Taking the square root of both sides, we find:
r = ±i√12 = ±2i√3
Since we have complex roots, the general solution of the differential equation will be composed of sine and cosine functions. According to the theory of homogeneous second-order differential equations, the general solution is given by:
y(t) = C_1 cos(2√3 t) + C_2 sin(2√3 t)
where C_1 and C_2 are constants determined by initial conditions or boundary conditions if provided. Thus, the general solution for the differential equation y” + 12y = 0 is:
y(t) = C_1 cos(2√3 t) + C_2 sin(2√3 t)
This solution reflects the oscillatory nature of the system described by the differential equation, characteristic of second-order linear equations with constant coefficients where the roots are purely imaginary.