Final answer:
To find the general solution for the differential equation (dy²)/(d²x)-4(dy)/(dx)+5y=0, use the characteristic equation method. The general solution is y = c₁e^(2x)cos(x) + c₂e^(2x)sin(x).
Step-by-step explanation:
To find the general solution for the differential equation (dy²)/(d²x)-4(dy)/(dx)+5y=0, we can use the characteristic equation method. This involves assuming a solution of the form y = e^(rx) and substituting it into the differential equation. By solving the resulting quadratic equation for r, we can determine the general solution.
Plugging in the values for the coefficients, we have r²-4r+5=0. Using the quadratic formula, we find the roots to be r=2±i, where i is the imaginary unit.
Hence, the general solution for the differential equation is y = c₁e^(2x)cos(x) + c₂e^(2x)sin(x), where c₁ and c₂ are arbitrary constants.