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The Differential Equation of the System of Curves y ax^2b cos nx
The Differential Equation of the System of Curves y ax^2 bcos(nx)
Exploring the mathematical underpinnings of systems of curves such as those defined by the equation y ax^2 bcos(nx) requires a thorough understanding of differential equations. By examining the functions and their derivatives, we can derive the differential equation that governs this system.
Deriving the Functions for the System of Curves
Let's consider the system defined by the equation y ax^2 bcos(nx), where a, b are constants and n is a fixed constant. We aim to find the differential equation that characterizes this system. To do this, we need to manipulate the expressions for the function and its derivatives.
Step 1: Define the Functions
Starting with the function y ax^2 bcos(nx), we define:
U ax^2 bcos(nx) U' 2ax - nbsin(nx) U'' 2a - n^2b cos(nx)Here, we have:
U represents the function value. U' represents the first derivative of the function. U'' represents the second derivative of the function.Step 2: Formulate the Determinant Condition
A necessary condition for the system to admit solutions for a and b is that the determinant of the matrix formed by the values of U, U', and U'' must vanish. This can be expressed as:
left| begin{array}{ccc} x^2 cos(nx) U 2x -nsin(nx) U' 2 -n^2cos(nx) U'' end{array} right| 0
This determinant condition encapsulates the relationship between the function values and their derivatives, ensuring that the system of curves is consistent and solvable for the given parameters a and b.
Step 3: Derive the Differential Equation
To derive the differential equation, we use the determinant condition:
[ begin{vmatrix} 2cos(nx) nxcos(nx) U'' 2x -nsin(nx) U' 2 -n^2cos(nx) U' end{vmatrix} 0 ]
We can expand this determinant to yield:
[ [2cos(nx) cdot (-nsin(nx) cdot U'') - nxcos(nx) cdot (2 cdot U'') 2 cdot (-nsin(nx) cdot U'')] - (2x cdot (-n^2cos(nx) cdot U') - 2 cdot (2 cdot U') 2 cdot (2 cdot (-n^2cos(nx) cdot U'')) 0 ]
Simplifying this expression, we obtain:
[ -2n^2cos(nx)U' - 2n^2cos(nx)U 0 ]
This simplifies further to:
[ (2cos(nx) nxcos(nx)U' - 2x cdot n^2cos(nx)U' - 2n^2cos(nx)U 0 ]
Further simplification leads to:
[ [2cos(nx) nxcos(nx)U' - xn^2cos(nx)U' 2n^2cos(nx)U 0 ]
Conclusion
The differential equation derived here encapsulates the behavior of the system of curves defined by U ax^2 bcos(nx). This equation provides a powerful tool for understanding and predicting the behavior of the system under various conditions. By analyzing the differential equation, we can gain deeper insights into the mathematical modeling of such systems, which can be applied in fields ranging from physics to engineering.