user: If I solve a differential equation and get complex conjugate solutions, how can I re-write them as trig functions instead of exp functions? [Timestamp: 2024/12/02 19:44:02]
ai: When you have complex conjugate solutions from a differential equation, they typically take the form:
[ y(t) = e^{\alpha t} (\cos(\beta t) + i \sin(\beta t)) ]
and its complex conjugate:
[ y(t) = e^{\alpha t} (\cos(\beta t) - i \sin(\beta t)) ]
where (\alpha) and (\beta) are real numbers. To rewrite these solutions in terms of trigonometric functions, you can use Euler’s formula:
[ e^{i\theta} = \cos(\theta) + i \sin(\theta) ]
For complex conjugate solutions, you can express them as:
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Combine the two solutions:
[ y(t) = e^{\alpha t} \cos(\beta t) + i e^{\alpha t} \sin(\beta t) ] [ y(t) = e^{\alpha t} \cos(\beta t) - i e^{\alpha t} \sin(\beta t) ]
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If you are interested in the real part, you can write:
[ y(t) = e^{\alpha t} \cos(\beta t) ]
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If you want to express the general solution, you can combine the real and imaginary parts:
[ y(t) = C_1 e^{\alpha t} \cos(\beta t) + C_2 e^{\alpha t} \sin(\beta t) ]
where (C_1) and (C_2) are constants determined by initial conditions.
This way, you can represent the complex solutions using trigonometric functions. [Timestamp: 2024/12/02 19:44:07]
user: If I have solutions of the form e^kx, how do you re-write that? [Timestamp: 2024/12/02 19:47:39]
ai: If you have solutions of the form ( e^{kx} ), where ( k ) is a constant, you can rewrite them in terms of trigonometric functions if ( k ) is complex.
Assuming ( k ) can be expressed as ( k = \alpha + i\beta ) (where ( \alpha ) and ( \beta ) are real numbers), you can use Euler’s formula:
[ e^{kx} = e^{(\alpha + i\beta)x} = e^{\alpha x} e^{i\beta x} = e^{\alpha x} (\cos(\beta x) + i \sin(\beta x)) ]
So, the expression ( e^{kx} ) can be rewritten as:
[ e^{kx} = e^{\alpha x} \cos(\beta x) + i e^{\alpha x} \sin(\beta x) ]
If ( k ) is purely real (i.e., ( k = \alpha )), then the expression remains as ( e^{kx} ) since it cannot be rewritten in terms of trigonometric functions.
In summary, if ( k ) is complex, you can express ( e^{kx} ) in terms of exponential decay/growth and trigonometric functions. If ( k ) is real, it stays as ( e^{kx} ). [Timestamp: 2024/12/02 19:47:42]