Final answer:
To solve a differential equation with an auxiliary equation that has complex roots, determine the roots of the quadratic equation, then express the general solution using sines and cosines with the roots’ coefficients. Apply initial conditions to derive specific constants if needed.
Step-by-step explanation:
When a differential equation has an auxiliary equation with complex roots, the general solution can be expressed using sines and cosines. To find the general solution, one would typically follow these steps:
- Solve the auxiliary equation, which usually takes the form of a quadratic equation, to find the roots.
- If the roots are complex, they will be in the form of α ± βi, where α and β are real numbers, and i is the imaginary unit.
- Write the general solution of the differential equation using the real part and the imaginary part of the roots as coefficients for the sine and cosine functions. That is, the general solution will be „x(t) = e^{αt}(C_1*cos(βt) + C_2*sin(βt))” where C_1 and C_2 are constants determined by initial conditions.
- Apply initial conditions if any are provided to find the particular solution.
In scenarios where multiple unknowns exist, additional equations or knowledge of physical principles may be necessary to derive the final answer. Understanding how to manipulate and solve equations is crucial, particularly when dealing with higher roots or complex solutions in equilibrium problems.