The rank of $$\left( {m \times n} \right)$$ matrix $$\left( {m < n} \right)$$ cannot be more then

The function $$y = {x^2} + {{250} \over x}$$ at $$x=5$$ attains

$$y = {e^{ - 2x}}$$ is a solution of the differential equation $$\,{y^{11}} + {y^1} - 2y = 0$$

Match each of the items $$A, B, C$$ with an appropriate item from $$1, 2, 3, 4$$ and $$5$$

List-$${\rm I}$$
$$(P)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}y{{dy} \over {dx}} + {a_3}y = {a_4}$$
$$(Q)$$ $${a_1}{{{d^2}y} \over {d{x^3}}} + {a_2}y = {a_3}$$
$$(Q)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}x{{dy} \over {dx}} + {a_3}{x^2}y = 0$$

List-$${\rm II}$$
$$(1)$$ Non-linear differential equation
$$(2)$$ Linear differential equation with constants coefficients
$$(3)$$ Linear homogeneous differential equation
$$(4)$$ Non-linear homogeneous differential equation
$$(5)$$ Non-linear first order differential equation