Closed form solution of a system of nonlinear differential equations

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Aleem Andrew 2020-2-4

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https://ww2.mathworks.cn/matlabcentral/answers/503665-closed-form-solution-of-a-system-of-nonlinear-differential-equations

评论： Walter Roberson 2020-2-5

The following code graphs the solution to a a system of nonlinear differential equations. How can I find the closed form solution to the system that expresses y as a function of x?

function dydt = odeproject(t,y,A,B)
if nargin < 3 || isempty(A)
  A = 1;    
end
if nargin < 4 || isempty(B)
  B = 2;
end
tspan = [0 0.5];
y0 = [2 7.524];
[t,y] = ode45(@(t,y) odefcn(t,y,A,B), tspan, y0);
plot(t,y(:,1),'-o',t,y(:,2),'-.')
end
function dydt = odefcn(t,y,A,B)
dydt = zeros(2,1);
dydt(1) = A(1);  
dydt(2) = B(1);  
dBdt = y(2) * sqrt((y(2)*y(2)+y(1)*y(1)));
dAdt = y(1) *  sqrt((y(2)*y(2)+y(1)*y(1))) -9.81;
end

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Aleem Andrew 2020-2-5

编辑：Aleem Andrew 2020-2-5

I am not sure how to write the code for such a function. I know how to create a function that has as parameters a dependent and an independent variable but not more than two. A and B are constants but should be the first derivatives of variables I need to differentiate twice with respect to a single independent variable. The code I wrote was intended to solve the following system of nonlinear differential equations:

x''=x'*sqrt(x'^2+y'^2) y''=y'*sqrt(x'^2+y'^2)-9.81

Walter Roberson 2020-2-5

在 MATLAB Online 中打开

You can do a change of variables to rewrite as

 XP' = XP * sqrt(XP^2 +. YP^2)
 YP' = YP * sqrt(XP^2 +. YP^2) - 9.81

and then run that as a system of two variables. If you need x and y you can do numeric integration such as cumtrapz. If you need a more accurate x and y you would use a system with four parameters.

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