def _linear_2eq_order1_type3(x, y, t, r, eq):
 (C1, C2, C3, C4) = get_numbered_constants(eq, num=4)
 F = Integral(r['a'], t)
 G = Integral(r['b'], t)
 sol1 = (exp(F) * ((C1 * exp(G)) + (C2 * exp((- G)))))
 sol2 = (exp(F) * ((C1 * exp(G)) - (C2 * exp((- G)))))
 return [Eq(x(t), sol1), Eq(y(t), sol2)]

def _linear_2eq_order1_type3 ( x , y , t , r , eq ) : ( C1 , C2 , C3 , C4 ) = get_numbered_constants ( eq , num = 4 ) F = Integral ( r [ 'a' ] , t ) G = Integral ( r [ 'b' ] , t ) sol1 = ( exp ( F ) * ( ( C1 * exp ( G ) ) + ( C2 * exp ( ( - G ) ) ) ) ) sol2 = ( exp ( F ) * ( ( C1 * exp ( G ) ) - ( C2 * exp ( ( - G ) ) ) ) ) return [ Eq ( x ( t ) , sol1 ) , Eq ( y ( t ) , sol2 ) ]
the equations of this type of ode are .