Not the right result for dsolve

Any kind of guidance would be much appreciated. I need to show that the exact solution of this ODE is y(t) = t tan(lnt).

syms y(t)
ode = diff(y,t) == 1+(y/t)+(y/t)^2;
cond = y(1) == 0;
ySol(t) = dsolve(ode,cond)

However, I keep getting

ySol(t) =

- t*1i - (2*t)/(t^2i*1i + 1i)

What am I doing wrong? Or am I missing something?

Answer 1

[Anay]

Hi Adan,

When solving the ODE using “dsolve”, MATLAB may return a complex solution due to ambiguity from inverse functions (e.g. arctan) and arbitrary constants during symbolic integration.

You can consider substituting “y(t)=t⋅u(t)” which will reduce the ode to a separable form, du/dt = (1 + u^2)/t. This change allows “dsolve” to solve the ODE without any ambiguity and gets the correct solution. You can use the below code for reference:

syms u(t)
% Let y = t*u
y = t*u;
dy = diff(y,t);

% dy = u + t*diff(u)
ode = dy == 1 + u + u^2;
ode_sub = ode;

% Solve
uSol(t) = dsolve(ode_sub, u(1) == 0);  % since y = t*u and y(1) = 0 -> u(1) = 0
ySol(t) = t*uSol(t)

ySol(t) = 

You can refer to the MATLAB documentation for more details about the dsolve function by following the below link:

https://www.mathworks.com/help/symbolic/dsolve.html