Solving a System of 2nd Order Linear ODE? "Explicit solutions cannot be found"

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Praxis Bays 2017-4-26

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https://ww2.mathworks.cn/matlabcentral/answers/337490-solving-a-system-of-2nd-order-linear-ode-explicit-solutions-cannot-be-found

评论： John D'Errico 2017-4-29

I've been trying to solve the linear 2nd order system below, but Matlab keeps saying there isn't a solution. Clearly, there's a solution because the solution will be the intersection of two sine and cosine waves of the same amplitude and phase shift. Because of this, I'm assuming the error is coming from syntax instead of from the math. Can someone take a look at it? Thanks in advanced.

This is the systems of equations I'm trying to solve:

x'' = G*M*sin(theta + 2*pi*t)/(x^2 + y^2)

y'' = G*M*cos(theta + 2*pi*t)/(x^2 + y^2)

Here's the code:

% variables
G = 6.674 * (10.^(-11));
M = 1.9886 * (10.^30); 
t = 0:.005:.7; 
xi = .44503 * 149.598 * 10.^9; % Initial x dist (AU --> m)
yi = .88106 * 149.598 * 10.^9; % Initial y dist (AU --> m)
thetai = atan((yi/xi)); % Initial angle from x-axis
% Calculations
syms x(t) y(t)
odex = diff(x) == diff(((G*M)./(x.^2 + y.^2)) .* cos(thetai + (2*pi).*t));
odey = diff(y) == diff(((G*M)./(x.^2 + y.^2)) .* sin(thetai + (2*pi).*t));
odes = [odex; odey]
S = dsolve(odes) % Returns not solvable here

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John D'Errico 2017-4-29

You misunderstand what MATLAB is telling you. That dsolve says there is no solution merely means there is no analytical solution. There may well be a solution. But an analytical solution, in terms of radicals, etc., no. There are many problems for which this is true, and they are easy to find.

So just use a numerical ODE solver, like ode45.

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Answer 1

Karan Gill 2017-4-29

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Since you're starting with floating point numbers, you should try a numerical simulation using ode45 or other numeric ODE solver. If you start numeric, stay numeric. Further, since a closed form symbolic solution couldn't be found, numeric is your remaining option.

Also, you declared "t" as a vector of numbers. But when "x(t)" and "y(t)" are defined then "t" gets overwritten by the symbolic variable "t".