dsolve hangs

[proy87]

Looks like a simple DE
y=Function('y')(x)
dsolve(diff(y,x,x)-8*cos(x)+sec(x)**2)

[sidhantnagpal]

This issue seems to have been fixed on master:

In [29]: y = Function('y')(x)                                                     

In [30]: dsolve(diff(y, x, x) - 8*cos(x) + sec(x)**2)                                 
Out[30]: y(x) = C₁ + C₂⋅x + log(cos(x)) - 8⋅cos(x)

Perhaps @oscarbenjamin will know if there's a need to add a test for this, if it wasn't already.

[oscarbenjamin]

I've checked and it is slow in 1.3 (doesn't hang) but it's slow in integrals. It faster in master because it is now solved by nth_algebraic:

In [8]: y=Function('y')(x) 
   ...: eq=diff(y,x,x)-8*cos(x)+sec(x)**2                                                                                                                     

In [9]: classify_ode(eq)                                                                                                                                      
Out[9]: 
('nth_algebraic',
 'nth_linear_constant_coeff_variation_of_parameters',
 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
 'nth_algebraic_Integral',
 'nth_linear_constant_coeff_variation_of_parameters_Integral',
 'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral')

The other solvers don't hang but they are slow and give suboptimal results:

In [1]: y=Function('y')(x) 
   ...: eq=diff(y,x,x)-8*cos(x)+sec(x)**2                                                                                                                     

In [2]: dsolve(eq, hint='nth_linear_euler_eq_nonhomogeneous_variation_of_parameters')                                                                         
Out[2]: 
                                           ⌠                          
                                           ⎮   ⎛              2   ⎞   
y(x) = C₁ + C₂⋅x + 8⋅x⋅sin(x) - x⋅tan(x) - ⎮ x⋅⎝8⋅cos(x) - sec (x)⎠ dx
                                           ⌡                          

In [3]: dsolve(eq, hint='nth_linear_constant_coeff_variation_of_parameters')                                                                                  
Out[3]: 
                                                           ⌠              
              ⎛                sin(x)⎞   ⌠                 ⎮       2      
y(x) = C₁ + x⋅⎜C₂ + 8⋅sin(x) - ──────⎟ - ⎮ 8⋅x⋅cos(x) dx - ⎮ -x⋅sec (x) dx
              ⎝                cos(x)⎠   ⌡                 ⌡  

This is because each method ends up with an integral like:

In [11]: integrate(x*(cos(x)-sec(x)**2), x)                                                                                                                   
Out[11]: 
⌠                        
⎮   ⎛            2   ⎞   
⎮ x⋅⎝cos(x) - sec (x)⎠ dx
⌡                        

which takes a while and doesn't evaluate anyway.

It could be worth opening a separate issue in integrals about that but I think this is all fixed as far as dsolve is concerned. Simple ODEs like this are now handled in a simple way by nth_algebraic.