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Lightfuzz
- 15
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How, if possible, could I solve the equation: x''x=((x')^2)/2? Thanks.
filiplarsen said:If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant.
A second order nonlinear ordinary differential equation is a mathematical equation that relates a function to its derivatives, where the highest derivative is of order two, and the equation is nonlinear, meaning that the function is not proportional to its derivatives. It is commonly used in physics and engineering to model complex systems.
In a linear differential equation, the function and its derivatives are only multiplied by constants, whereas in a nonlinear differential equation, they may also be multiplied by variables or functions. This results in a more complex relationship between the function and its derivatives, making it more challenging to solve analytically.
Second order nonlinear ordinary differential equations are used to model a wide range of phenomena in the physical and natural sciences, including population growth, chemical reactions, fluid dynamics, and electrical circuits. They are also used in economics, finance, and other fields to analyze complex systems and predict future behavior.
There is no general method for solving all second order nonlinear ordinary differential equations, but there are several techniques that can be used depending on the specific equation. These include separation of variables, substitution, and series solutions. In some cases, numerical methods must be used to approximate a solution.
While second order nonlinear ordinary differential equations are powerful tools for modeling complex systems, they have some limitations. They may not accurately represent a system if the underlying assumptions are incorrect, and they may be difficult or impossible to solve analytically. Additionally, small changes in the initial conditions or parameters of the equation can result in significantly different solutions, making it challenging to predict long-term behavior.