- #1
Logik
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I have to solve an ODE with variation of coefficient technique. It's pretty easy but I have no clue what is the first and second derivative of e^ix and e^-ix.
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Logik said:e^ix
first
i*e^ix
second
i^2*e^ix
e^-ix
first
-i*e^-ix
second
i^2*e^-ix
That is exactly what has been answered in each of these responses. For any constant, a, the derivative of [itex]e^{ax}[/itex] is [itex]ae^{ax}[/itex].bgbaby said:I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
bgbaby said:I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
The Variation of Coefficient Technique is a method used to solve Ordinary Differential Equations (ODEs) with complex derivatives. It involves rewriting the ODE as a product of a linear operator and a complex-valued function, and then using the properties of complex numbers to solve the resulting equation.
The Variation of Coefficient Technique is particularly useful for solving ODEs with complex derivatives, as it allows for the use of complex numbers and their properties to simplify the equations and find solutions. It can also be used for higher order ODEs, as it reduces them to first order equations.
To apply the Variation of Coefficient Technique, the ODE is first rewritten in the form of a linear operator acting on a complex-valued function. The operator is then factored into its eigenvalues and eigenvectors, and the solution is expressed in terms of these eigenvectors and a time-dependent complex coefficient.
The Variation of Coefficient Technique allows for the use of complex numbers, which can simplify the equations and provide more elegant solutions. It also reduces higher order ODEs to first order equations, making them easier to solve. Additionally, it can be used to solve both homogeneous and non-homogeneous ODEs.
The Variation of Coefficient Technique may not always be the most efficient method for solving ODEs, as it can be more complex and time-consuming than other techniques. It also requires a good understanding of complex numbers and their properties. It may not be suitable for all types of ODEs, as some equations may not easily fit into the required form for this technique.