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peace-Econ
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Homework Statement
Find a second order differential ewuation for which three functions y=2e^-t, y=2te^-t, y=e^(-t+1) are solutions.
If those functions were independent, this would be impossible but [itex]e^{-t+ 1}= e^{-t}e^1= e e^{-t}[/itex], a constant times [itex]e^{-t}[/itex] so you really have only two independent solutions.peace-Econ said:Homework Statement
Find a second order differential ewuation for which three functions y=2e^-t, y=2te^-t, y=e^(-t+1) are solutions.
Homework Equations
The Attempt at a Solution
A second order differential equation is an equation that contains a second derivative of an unknown function. It is typically written in the form of y'' = f(x,y,y'), where y' is the first derivative and y'' is the second derivative.
A linear second order differential equation has the form y'' + p(x)y' + q(x)y = g(x), where p(x) and q(x) are functions of x and g(x) is a function of x. A nonlinear second order differential equation is any equation that cannot be written in this form.
The method for solving a second order differential equation depends on the type of equation. For linear equations, it can be solved using techniques such as separation of variables, integrating factors, or series solutions. Nonlinear equations may require numerical methods or approximation techniques to find solutions.
Second order differential equations are used to model many physical phenomena, such as motion, oscillations, and electrical circuits. They are also commonly used in engineering, economics, and other fields to analyze and predict behavior of systems.
Yes, a second order differential equation can have multiple solutions. In fact, the general solution to a second order differential equation will typically include two arbitrary constants, allowing for an infinite number of solutions.