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ironcross77
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why y=e^mx is alwayas taken as trial soln in solving 2nd order diff equations. please explain in details.
ironcross77 said:why y=e^mx is alwayas taken as trial soln in solving 2nd order diff equations. please explain in details.
Actually only for linear equations with constant coefficients.matt grime said:You don't do that for *all* DE's, only linear ones, and only to get the homogeneous solutions.
The fundamental theorem of algebra says that any nth degree polynomial has exactly n complex roots (including multilicity).
ironcross77 said:What do u exactly mean by the above? Say a third degree polynomial does not have exactly 3 complex roots. Complex roots always occur in conjugate pairs ie. in any polynomial complex roots must occur in even nos.Code:The fundamental theorem of algebra says that any nth degree polynomial has exactly n complex roots (including multilicity).
What d_leet said pretty much answers the question, but I also just wanted to point out that complex roots do not always come in conjugate pairs. This is the case for polynomials with real coefficients, but not true when complex coefficients are allowed. For example the polynomial you get by expanding (x-(1+i))(x-1)2 would clearly have 1+i as a root but not 1-i. That said, I agree with you that my wording was somewhat misleading. A better way to phrase it would be "has exactly n roots in the field of complex numbers".ironcross77 said:What do u exactly mean by the above? Say a third degree polynomial does not have exactly 3 complex roots. Complex roots always occur in conjugate pairs ie. in any polynomial complex roots must occur in even nos.Code:The fundamental theorem of algebra says that any nth degree polynomial has exactly n complex roots (including multilicity).
There are a few reasons why y=e^mx is a popular choice as a trial solution in scientific experiments. Firstly, it is a simple and well-known function that is easy to work with mathematically. Additionally, it has important properties such as being differentiable and having a constant non-zero derivative, making it versatile for a wide range of applications. Finally, it is a good approximation for many real-world phenomena, making it a practical choice for modeling data.
Yes, other functions can be used as trial solutions, depending on the specific experiment and the problem at hand. However, y=e^mx is often the preferred choice due to its simplicity and the reasons mentioned in the previous answer. Other functions may also be used when they are more suitable for the particular experiment, such as trigonometric functions for modeling periodic data.
In solving differential equations, y=e^mx is used as a trial solution to substitute into the equation and find a particular solution. This is known as the method of undetermined coefficients. By plugging in the trial solution and its derivatives, the coefficients can be determined and a particular solution can be found. This method is commonly used in differential equations involving exponential growth or decay.
While y=e^mx is a useful and versatile function, it does have some limitations when used as a trial solution. Firstly, it can only approximate certain types of functions, so it may not be suitable for all types of data. Additionally, it may not be the best choice for problems involving non-linear relationships. Other functions, such as polynomials, may be better suited for these types of problems.
The best way to determine if y=e^mx is a good fit for your data is to plot the function and compare it to your data points. If the curve of the function closely follows the trend of your data, then it is likely a good fit. Additionally, you can use statistical methods such as regression analysis to evaluate the fit of the function to your data. It is also important to consider the limitations of using y=e^mx as mentioned in the previous answer when determining its suitability for your data.