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wofsy
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Can someone tell me how to solve the ODE,
d2x/dt2 = -cosx d2y/dt2 = -cosy in the plane?
d2x/dt2 = -cosx d2y/dt2 = -cosy in the plane?
HallsofIvy said:Those two equations are completely independent- so it is not necessary to "separate variables". You are just solving two separate second order differential equations. And, in fact you just need to integrate each twice.
HallsofIvy said:If d2x/dt2= cos(x) is a single equation in the dependent variable x as a function of t. There is absolutely no reason to introduce y. Since the independent variable "t" does not appear in the equation, I would use "quadrature":
Let v= dx/dt so that d2x/dt2= dv/dt= (dv/dx)(dx/dt)= v dv/dx. Now you have vdv/dx= cos(x) or vdv= cos(x)dx. Integrating, (1/2)v2= sin(x)+ C. dx/dt= v= [itex]\sqrt{2(sin(x)+ C)}[/itex] or
[tex]\frac{dx}{\sqrt{2(sin(x)+ C)}}= dt[/tex]
That left side is an "elliptical integral".
Of course, y will be exactly the same, though possibly with different constants of integration.
An ODE (ordinary differential equation) in plane refers to a differential equation that involves two independent variables, usually represented as x and y. The equation can be solved to find the values of x and y that satisfy the equation.
This notation means that the second derivative of x with respect to time (t) is equal to the cosine of x. In other words, the rate of change of the rate of change of x is equal to the cosine of x.
To solve these equations, we need to use techniques from calculus, such as integration and differentiation. We can also use numerical methods or computer software to find approximate solutions.
The possible solutions depend on the initial conditions given for x and y. These initial conditions determine the specific values of x and y that satisfy the equations. There can be multiple solutions or no solution at all.
Solving ODEs in plane is important in science because many natural phenomena can be described by these equations. From predicting the motion of planets to modeling chemical reactions, ODEs in plane help us understand and make predictions about the world around us.