DO3

Solve $(2D^2 + 5D - 12)y = 0\,$

Let $f(D) = 2D^2 + 5D - 12\,$

Find the roots of the characteristic equation $f(m) = 2m^2 + 5m - 12\,$

$f(m) = (m+4)(2m-3)\,$

Since $f(D)e^{mx} = e^{mx}f(m)\,$ , if $f(m) = 0\,$ then $f(D)e^{mx} = 0\,$ .

The roots are $m=-4,\frac{3}{2}\,$

So the problem reduces to

$(2D^2 + 5D - 12)e^{-4x}=0\,$

$(2D^2 + 5D - 12)e^{\frac{3}{2}x}=0\,$

So two solutions are $y_1 = e^{-4x}\,$ and $y_2 = e^{\frac{3}{2}x}\,$

Hence the general solution is $y = Ae^{-4x} + Be^{\frac{3}{2}x} \,$