Question

Solve the for the general solution of the differential equation (D^{6}+6D^{4}+12D^{2}+8)y=0

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ANSWERED
asked 2021-05-12
Solve the for the general solution of the differential equation
\(\displaystyle{\left({D}^{{{6}}}+{6}{D}^{{{4}}}+{12}{D}^{{{2}}}+{8}\right)}{y}={0}\)

Expert Answers (1)

2021-05-14

Step 1
Given differential equation is
\(\displaystyle{\left({D}^{{{6}}}+{6}{D}^{{{4}}}+{12}{D}^{{{2}}}+{8}\right)}{y}={0}\text{ where }{D}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}\)
Auxiliary equation is \(\displaystyle{m}^{{{6}}}+{6}{m}^{{{4}}}+{12}{m}^{{{2}}}+{8}={0}\)
Step 2
\(\displaystyle{\left({m}^{{{2}}}+{2}\right)}^{{{3}}}={0}\ \ \ {\left(\because{\left({a}+{b}\right)}^{{{3}}}={a}^{{{3}}}+{3}{a}^{{{2}}}{b}+{3}{a}{b}^{{{2}}}+{b}^{{{3}}}\right)}\)
\(\displaystyle{m}^{{{2}}}=-{2},-{2},-{2}\)
\(\displaystyle{m}=\pm\sqrt{{{2}}}{i},\pm\sqrt{{{2}}}{i},\pm\sqrt{{{2}}}{i}\)
\(\displaystyle\because{y}={\left({c}_{{{1}}}{x}^{{{2}}}+{c}_{{{2}}}{x}+{c}_{{{3}}}\right)}{\cos{{\left(\sqrt{{{2}}}{x}\right)}}}+{\left({c}_{{{4}}}{x}^{{{2}}}+{c}_{{{5}}}{x}+{c}_{{{6}}}\right)}{\sin{{\left(\sqrt{{{2}}}{x}\right)}}}\)

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