In what range should $$Re(s)$$ remain so that the laplace transform of the function $${e^{\left( {a + 2} \right)t + 5}}$$ exists?

Match the following and choose the correct combination

Group $$-$$ $${\rm I}$$
$$E.$$ Newton $$-$$ Raphson method
$$F.$$ Runge-Kutta method
$$G.$$ Simpson's Rule
$$H.$$ Gauss elimination

Group $$-$$ $${\rm II}$$
$$(1)$$ Solving non-linear equations
$$(2)$$ Solving linear simultaneous equations
$$(3)$$ Solving ordinary differential equations
$$(4)$$ Numerical integration method
$$(5)$$ Interpolation
$$(6)$$ Calculation of eigen values

The following differential equation has $$3{{{d^2}y} \over {d{t^2}}} + 4{\left( {{{dy} \over {dt}}} \right)^3} + {y^2} + 2 = x$$

A solution of the differential equation $${{{d^2}y} \over {d{x^2}}} - 5{{dy} \over {dx}} + 6y = 0\,$$ is given by