Question
Ask by Graham O'Quinn. in China
Jan 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The general solution to the differential equation is:
where
and
are constants.
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The Deep Dive
To solve the given differential equation, we begin by finding the characteristic equation by substituting
. This leads to:
Using the quadratic formula,
, we substitute
,
, and
:
Thus, the general solution is:
where
and
are constants determined by initial conditions.
The equation you have here is a second-order linear homogeneous differential equation with constant coefficients, and it arises in various fields such as physics and engineering, especially in systems involving damped oscillations. In practice, these equations can model real-world phenomena like the motion of a spring or a circuit response in electronics.
To avoid common mistakes when solving these types of equations, always double-check your substitution and ensure you understand the complex roots. Remember that these solutions often involve combinations of exponential, sine, and cosine functions, and correctly interpreting the real part (growth/decay) and the imaginary part (oscillation) is key!