Solve nonlinear 2nd order ODE numerically

Question

Lucas 2022-7-28

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1769845-solve-nonlinear-2nd-order-ode-numerically

评论： MOSLI KARIM 2022-8-12

采纳的回答： Sam Chak

I need to solve the following nonlinear 2nd order ODE, that is, find

such that

$1-x=-\frac{y''(x)}{(1+(y'(x))^2)^{{3/2}}$

I tried using

>> syms y(x)

>> ode = -diff(y,x,2)/(1+(diff(y,x))^2)^(3/2) == 1-x;

>> ySol(x) = dsolve(ode)

but it doesn't work since apparently there is no anaylitical solution (if I rearrange the terms it does find a system of complex solutions, but I think the it is not right).

Isn't there a command to solve ODEs numerically? I am expeting something like the family of plots from here https://www.wolframalpha.com/input?i=f%27%27%28t%29%2F%28%281%2B%28f%27%28t%29%29%5E2%29%5E%283%2F2%29%29+%3D+-%281-0.25t%29

Many thanks oin advance!

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Torsten 2022-7-28

What are your initial/boundary conditions for y ?

Lucas 2022-7-29

My idea was to screen these conditions to find one that satisfies my problem.

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Answer 1

Sam Chak 2022-7-28

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1769845-solve-nonlinear-2nd-order-ode-numerically#answer_1017055

在 MATLAB Online 中打开

Hi @Lucas

You can follow the example here

https://www.mathworks.com/help/matlab/ref/ode45.html#bu3uhuk

and try something like this:

tspan = [0 1.15];

y0 = [1 0]; % initial condition

[t,y] = ode45(@(t, y) odefcn(t, y), tspan, y0);

plot(t, y(:,1)), grid on, xlabel('t')

function dydt = odefcn(t, y)

dydt = zeros(2,1);

c = 0.25;

dydt(1) = y(2);

dydt(2) = - (1 - c*t)*(1 + y(2)^2)^(3/2);

end

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Lucas 2022-7-29

That worked, thanks!

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Answer 2

James Tursa 2022-7-28

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1769845-solve-nonlinear-2nd-order-ode-numerically#answer_1016970

You can look at the examples for ode45( ) here:

https://www.mathworks.com/help/matlab/ref/ode45.html?searchHighlight=ode45&s_tid=srchtitle_ode45_1

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Answer 3

MOSLI KARIM 2022-8-12

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1769845-solve-nonlinear-2nd-order-ode-numerically#answer_1025340

在 MATLAB Online 中打开

function  pvb_pr13
tspan=[0 1.5];
y0=[1 0];
[x,y]=ode45(@fct,tspan,y0);
figure(1)
hold on 
plot(x,y(:,1),'r-')
grid on 
    function yp=fct(x,y)
    c=0.25;
      yp=[y(2);-(1-c*x)*((1+(y(2))^2)^(3/2))];  
end 
end 