Final answer:
To solve the linear differential equation y³-y²+y-y=0, we can use the method of separation of variables. The general solution to the differential equation is y(x) = c₁e^0x + c₂e^((1/2 + i√3/2)x) + c₃e^((1/2 - i√3/2)x).
Step-by-step explanation:
To solve the linear differential equation y³-y²+y-y=0, we can use the method of separation of variables.
Rearranging the equation, we have y³-y²+y-y = 0.
which can be factored as y(y²-y+1) = 0.
Setting each factor equal to zero, we find y = 0 and y²-y+1 = 0.
The quadratic equation y²-y+1 = 0 can be solved using the quadratic formula, and
We find two complex solutions: y = 1/2 + i√3/2 and y = 1/2 - i√3/2.
Therefore, the general solution to the differential equation is y(x) = c₁e^(0x) + c₂e^((1/2 + i√3/2)x) + c₃e^((1/2 - i√3/2)x), where c₁, c₂, and c₃ are constants.