The Method of Averaging

x' (t)  =  
        e × (cos^2) t  × \[RawEscape]x (t)

e = 1/100;

sol1 = 
        DSolve[ {x[0] == 1, x'[t] ==   + e x[t] Cos[t]^2}, x, 
          t];

\(\(sol2 = 
        DSolve[\ {x[0] == 1, \(x'\)[t] == \ \ \(+\ e\)\ x[t]\ 1/2}, x, 
          t];\)

\)
Plot[{ x[t]  /.  sol1,   x[t]  /.  sol2,  
        1 +  e Cos[t]^2} ,  {t, 0, 40}]

sol1 = 
        DSolve[ {x[0] == 1, x'[t] ==   - e x[t] Cos[t]^2}, x, 
          t];

\(\(sol2 = 
        DSolve[\ {x[0] == 1, \(x'\)[t] == \ \ \(-\ e\)\ x[t]\ 1/2}, x, 
          t];\)

\)
Plot[{ x[t]  /.  sol1,  e x[t] Cos[t]^2  /.  sol1,  
        x[t]  /.  sol2} ,  {t, 0, 40}]

Plot[{ x[t]  /.  sol1,  
        e x[t] Cos[t]^2  /.  sol1} ,  {t, 0, 40}]

Plot[{ x[t]  /.  sol1,  
        e x [ t] Cos[t]^2  /.  sol1,  
        e x[2 Pi Floor[t/(2  Pi)]] Cos[t]^2  /.  sol1} ,  {t, 0, 
        40}]