Given: \(\displaystyle{y}\text{}{4}{y}={0}\)

\(\displaystyle\rightarrow{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}-{4}{y}={0}\)

Let \(\displaystyle{\frac{{{d}^{{{2}}}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={0}^{{{2}}}=\) differential operator

\(\displaystyle{D}^{{{2}}}{y}-{4}{y}={0}\)

\(\displaystyle{\left({D}^{{{2}}}-{4}\right)}{y}={0}\)

For complementary sol put \(\displaystyle{D}^{{{2}}}-{4}={0}\)

\(\displaystyle{D}^{{{2}}}-{4}={0}\)

\(\displaystyle{D}^{{{2}}}={4}\)

\(\displaystyle{D}=\pm{2}\)

\(\displaystyle{m}_{{{1}}}={2}\ \text{and}\ {m}_{{{2}}}=-{2}\)

\(\displaystyle{y}={c}_{{{1}}}{e}^{{{m}{x}}}+{c}_{{{2}}}{e}^{{{m}{2}{x}}}\)

\(\displaystyle{c}_{{{1}}}\ \text{and}\ {c}_{{{2}}}=\text{constants}\)

Answer: \(\displaystyle{y}={c}_{{{1}}}{e}^{{{2}{x}}}+{c}_{{{2}}}{e}^{{-{2}{x}}}\)

\(\displaystyle\rightarrow{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}-{4}{y}={0}\)

Let \(\displaystyle{\frac{{{d}^{{{2}}}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={0}^{{{2}}}=\) differential operator

\(\displaystyle{D}^{{{2}}}{y}-{4}{y}={0}\)

\(\displaystyle{\left({D}^{{{2}}}-{4}\right)}{y}={0}\)

For complementary sol put \(\displaystyle{D}^{{{2}}}-{4}={0}\)

\(\displaystyle{D}^{{{2}}}-{4}={0}\)

\(\displaystyle{D}^{{{2}}}={4}\)

\(\displaystyle{D}=\pm{2}\)

\(\displaystyle{m}_{{{1}}}={2}\ \text{and}\ {m}_{{{2}}}=-{2}\)

\(\displaystyle{y}={c}_{{{1}}}{e}^{{{m}{x}}}+{c}_{{{2}}}{e}^{{{m}{2}{x}}}\)

\(\displaystyle{c}_{{{1}}}\ \text{and}\ {c}_{{{2}}}=\text{constants}\)

Answer: \(\displaystyle{y}={c}_{{{1}}}{e}^{{{2}{x}}}+{c}_{{{2}}}{e}^{{-{2}{x}}}\)