Why two methods give different results, and which is correct?

I use ode15i and ode45 to solve an implicit function, however two methods give different results, which is correct?

clear
clc
close all

%% ode15i
f = @(t,y,yp)yp.^2-5/(0.5*yp-2./y);
[t0,y0,yp0,tspan] = deal(0,1.924,2.079,[0 1]);
[y0,yp0] = decic(f,t0,y0,1,yp0,0);
sol = ode15i(f,tspan,y0,yp0);
subplot(1,2,1)
plot(sol.x,sol.y)
title('ode15i')

%% ode45
tspan = [0 1];                                                             % time interval
y0 = 1.924;                                                                % initial value

[t,y] = ode45(@(t,y)odefcn(t,y),tspan,y0);                                 % ode45
subplot(1,2,2)
plot(t,y)
title('ode45')

% define function
function Dy = odefcn(t,y)
fun = @(Dy) Dy.^2 - 5/(0.5*Dy-2/y);                                        % implicit function
Dy = fzero( fun,0);                                                        % fzero function to solve implict function
end

and results are plotted below:

Answer 1

[Davide Masiello]

Since you have an implicit ODE, the correct solution is given by ode15i.

The problem with ode45 is in the initial guess used to find the value of Dy. I used 5 instead of 0 in the exmple below and got the same result of ode15i.

clear
clc
close all

%% ode15i
f = @(t,y,yp)yp.^2-5/(0.5*yp-2./y);
[t0,y0,yp0,tspan] = deal(0,1.924,2.079,[0 1]);
[y0,yp0] = decic(f,t0,y0,1,yp0,0);
sol = ode15i(f,tspan,y0,yp0);
subplot(1,2,1)
plot(sol.x,sol.y)
title('ode15i')

%% ode45
tspan = [0 1];                                                             % time interval
y0 = 1.924;                                                                % initial value
[t,y] = ode45(@(t,y)odefcn(t,y),tspan,y0);                                 % ode45
subplot(1,2,2)
plot(t,y)
title('ode45')

% define function
function Dy = odefcn(t,y)
fun = @(Dy) Dy.^2 - 5/(0.5*Dy-2/y);                                        % implicit function
Dy = fzero(fun,5);                                                        % fzero function to solve implict function
end

[warnerchang]

thank you so much!