Solve nonlinear 2nd order ODE numerically

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

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!

[Torsten]

What are your initial/boundary conditions for y ?

[Lucas]

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

Answer 1

[Sam Chak]

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

[Lucas]

That worked, thanks!

Answer 2

[James Tursa]

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

Answer 3

[MOSLI KARIM]

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 

[MOSLI KARIM]

you can used this code for solved your problem