(x, y, t, r)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
r"""
    Equations are '\\frac{d}{d t} x{\\left (t \\right )} = f{\\left (t \\right )} x{\\left (t \\right )} + ' \
    'g{\\left (t \\right )} y{\\left (t \\right )}' and
    '\\frac{d}{d t} y{\\left (t \\right )} = a g{\\left (t \\right )} x{\\left (t \\right )} + ' \
    '\\left(b g{\\left (t \\right )} + f{\\left (t \\right )}\\right) y{\\left (t \\right )}'
    The transformation of
    'x{\\left (t \\right )} = u{\\left (T \\right )} e^{\\int f{\\left (t \\right )}\\, dt}',
    'y{\\left (t \\right )} = v{\\left (T \\right )} e^{\\int f{\\left (t \\right )}\\, dt}' and
    'T{\\left (t \\right )} = e^{\\int g{\\left (t \\right )}\\, dt}' leads to a system of
    constant coefficient linear differential equations '\\frac{d}{d T} u{\\left (T \\right )} = v{\\left (T \\right )}'
    and '\\frac{d}{d T} v{\\left (T \\right )} = a u{\\left (T \\right )} + b v{\\left (T \\right )}'.
r"""
    Equations are '\\frac{d}{d t} x{\\left (t \\right )} = f{\\left (t \\right )} x{\\left (t \\right )} + ' \
    'g{\\left (t \\right )} y{\\left (t \\right )}' and
    '\\frac{d}{d t} y{\\left (t \\right )} = a g{\\left (t \\right )} x{\\left (t \\right )} + ' \
    '\\left(b g{\\left (t \\right )} + f{\\left (t \\right )}\\right) y{\\left (t \\right )}'
    The transformation of
    'x{\\left (t \\right )} = u{\\left (T \\right )} e^{\\int f{\\left (t \\right )}\\, dt}',
    'y{\\left (t \\right )} = v{\\left (T \\right )} e^{\\int f{\\left (t \\right )}\\, dt}' and
    'T{\\left (t \\right )} = e^{\\int g{\\left (t \\right )}\\, dt}' leads to a system of
    constant coefficient linear differential equations '\\frac{d}{d T} u{\\left (T \\right )} = v{\\left (T \\right )}'
    and '\\frac{d}{d T} v{\\left (T \\right )} = a u{\\left (T \\right )} + b v{\\left (T \\right )}'.
r Equations are \\ frac { d } { d t } x { \\ left ( t \\ right ) } = f { \\ left ( t \\ right ) } x { \\ left ( t \\ right ) } + \ g { \\ left ( t \\ right ) } y { \\ left ( t \\ right ) } and \\ frac { d } { d t } y { \\ left ( t \\ right ) } = a g { \\ left ( t \\ right ) } x { \\ left ( t \\ right ) } + \ \\ left ( b g { \\ left ( t \\ right ) } + f { \\ left ( t \\ right ) } \\ right ) y { \\ left ( t \\ right ) } The transformation of x { \\ left ( t \\ right ) } = u { \\ left ( T \\ right ) } e^ { \\ int f { \\ left ( t \\ right ) } \\ dt } y { \\ left ( t \\ right ) } = v { \\ left ( T \\ right ) } e^ { \\ int f { \\ left ( t \\ right ) } \\ dt } and T { \\ left ( t \\ right ) } = e^ { \\ int g { \\ left ( t \\ right ) } \\ dt } leads to a system of constant coefficient linear differential equations \\ frac { d } { d T } u { \\ left ( T \\ right ) } = v { \\ left ( T \\ right ) } and \\ frac { d } { d T } v { \\ left ( T \\ right ) } = a u { \\ left ( T \\ right ) } + b v { \\ left ( T \\ right ) } .
def _linear_2eq_order1_type5(x, y, t, r):
    r"""
    Equations are '\\frac{d}{d t} x{\\left (t \\right )} = f{\\left (t \\right )} x{\\left (t \\right )} + ' \
    'g{\\left (t \\right )} y{\\left (t \\right )}' and
    '\\frac{d}{d t} y{\\left (t \\right )} = a g{\\left (t \\right )} x{\\left (t \\right )} + ' \
    '\\left(b g{\\left (t \\right )} + f{\\left (t \\right )}\\right) y{\\left (t \\right )}'
    The transformation of
    'x{\\left (t \\right )} = u{\\left (T \\right )} e^{\\int f{\\left (t \\right )}\\, dt}',
    'y{\\left (t \\right )} = v{\\left (T \\right )} e^{\\int f{\\left (t \\right )}\\, dt}' and
    'T{\\left (t \\right )} = e^{\\int g{\\left (t \\right )}\\, dt}' leads to a system of
    constant coefficient linear differential equations '\\frac{d}{d T} u{\\left (T \\right )} = v{\\left (T \\right )}'
    and '\\frac{d}{d T} v{\\left (T \\right )} = a u{\\left (T \\right )} + b v{\\left (T \\right )}'.
    """
    C1, C2, C3, C4 = symbols('C1:5')
    u, v = symbols('u, v', function=True)
    T = Symbol('T')
    if not cancel(r['c']/r['b']).has(t):
        p = cancel(r['c']/r['b'])
        q = cancel((r['d']-r['a'])/r['b'])
        eq = (Eq(diff(u(T),T), v(T)), Eq(diff(v(T),T), p*u(T)+q*v(T)))
        sol = dsolve(eq)
        sol1 = exp(C.Integral(r['a'], t))*sol[0].rhs.subs(T, C.Integral(r['b'],t))
        sol2 = exp(C.Integral(r['a'], t))*sol[1].rhs.subs(T, C.Integral(r['b'],t))
    if not cancel(r['a']/r['d']).has(t):
        p = cancel(r['a']/r['d'])
        q = cancel((r['b']-r['c'])/r['d'])
        sol = dsolve(Eq(diff(u(T),T), v(T)), Eq(diff(v(T),T), p*u(T)+q*v(T)))
        sol1 = exp(C.Integral(r['c'], t))*sol[1].rhs.subs(T, C.Integral(r['d'],t))
        sol2 = exp(C.Integral(r['c'], t))*sol[0].rhs.subs(T, C.Integral(r['d'],t))
    return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order1_type5 ( x , y , t , r ) : C1 , C2 , C3 , C4 = symbols ( 'C1:5' ) u , v = symbols ( 'u, v' , function = True ) T = Symbol ( 'T' ) if not cancel ( r [ 'c' ] / r [ 'b' ] ) . has ( t ) : p = cancel ( r [ 'c' ] / r [ 'b' ] ) q = cancel ( ( r [ 'd' ] - r [ 'a' ] ) / r [ 'b' ] ) eq = ( Eq ( diff ( u ( T ) , T ) , v ( T ) ) , Eq ( diff ( v ( T ) , T ) , p * u ( T ) + q * v ( T ) ) ) sol = dsolve ( eq ) sol1 = exp ( C . Integral ( r [ 'a' ] , t ) ) * sol [ 0 ] . rhs . subs ( T , C . Integral ( r [ 'b' ] , t ) ) sol2 = exp ( C . Integral ( r [ 'a' ] , t ) ) * sol [ 1 ] . rhs . subs ( T , C . Integral ( r [ 'b' ] , t ) ) if not cancel ( r [ 'a' ] / r [ 'd' ] ) . has ( t ) : p = cancel ( r [ 'a' ] / r [ 'd' ] ) q = cancel ( ( r [ 'b' ] - r [ 'c' ] ) / r [ 'd' ] ) sol = dsolve ( Eq ( diff ( u ( T ) , T ) , v ( T ) ) , Eq ( diff ( v ( T ) , T ) , p * u ( T ) + q * v ( T ) ) ) sol1 = exp ( C . Integral ( r [ 'c' ] , t ) ) * sol [ 1 ] . rhs . subs ( T , C . Integral ( r [ 'd' ] , t ) ) sol2 = exp ( C . Integral ( r [ 'c' ] , t ) ) * sol [ 0 ] . rhs . subs ( T , C . Integral ( r [ 'd' ] , t ) ) return [ Eq ( x ( t ) , sol1 ) , Eq ( y ( t ) , sol2 ) ]