DO3

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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}}\,

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