You have to solve the non-linear time variant differential equation:

(cos(y)+2)*y''+atan(y)*sin(0.01*t)^2+y*sin(0.5*t)=0

with initial condition as input of the function

[y(0);y'(0)]=[a,b]

and return the time 'te' when 'y' crosses zero from the negatives values to the positive values.

te=zero_crossing(a,b)

tip: use the 'Events' option in odeset to detect the crossing, stop integration and get the time 'te'. See how to use it here : http://www.mathworks.com/help/matlab/math/ode-event-location.html?searchHighlight=ballode

Additional info:

• in the test case, the solver ode45 is used, with options 'RelTol'=1e-8 and 'AbsTol'=1e-10.
• because you don't know in advance the final time of the crossing, you can solve again further in time if no crossing is detected, starting from the point of the final time of previous integration.