To find the general solution of the higher order differential equation
y”” + 2y = 0,
we’ll follow a systematic approach.
Step 1: Write the Characteristic Equation
The first step in solving a linear homogeneous differential equation is to form the characteristic equation. For a differential equation of the form
y”” + ay = 0, the characteristic equation is obtained by replacing
y
with
r, leading us to:
r^4 + 2 = 0
Step 2: Solve the Characteristic Equation
Next, we solve the characteristic equation:
r^4 = -2
Taking the fourth root of both sides gives us:
r =
ext{±} rac{1}{
oot{4} imes ext{sqrt{2}}},
In complex form, the roots can be expressed as:
r = ± rac{1}{
oot{4} imes 2^{1/2}} imes (cos( rac{ rac{ ext{π}}{4}}{2}) + i imes sin( rac{ rac{ ext{π}}{4}}{2}))
Step 3: Write the General Solution
With the roots found, we can write the general solution. The general solution for a fourth-order differential equation with complex roots is given by:
y(t) = C_1 imes e^{ rac{ ext{π}}{4}} + C_2 imes e^{- rac{ ext{π}}{4}} + C_3 imes cos(t) + C_4 imes sin(t)
Step 4: Conclusion
Therefore, the general solution of the differential equation y”” + 2y = 0 is:
y(t) = C_1 imes e^{ rac{1}{
oot{2}} imes t} + C_2 imes e^{- rac{1}{
oot{2}} imes t} + C_3 imes cos( rac{t}{
oot{2}}) + C_4 imes sin( rac{t}{
oot{2}})
where C1, C2, C3, and C4 are arbitrary constants determined by boundary conditions.