solve the Differential equations y'''+10y''+25y'=0

solve the Differential equations y'''+10y''+25y'=0

Question
Differential equations
asked 2020-11-09
solve the Differential equations \(\displaystyle{y}{'''}+{10}{y}{''}+{25}{y}'={0}\)

Answers (1)

2020-11-10
The given Differential equations is,
\(\displaystyle{y}{'''}+{10}{y}{''}+{25}{y}′={0}.\)
This equation can be re-written as,
\(\displaystyle{\left({D}^{{{3}}}+{10}{D}^{{{2}}}+{25}{D}\right)}{y}={0},\) where \(\displaystyle{D}≡{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}.\)The auxiliary equation of (1) is, \(\displaystyle{D}^{{{3}}}+{10}{D}^{{{2}}}+{25}{D}={0}\)
\(\displaystyle\Rightarrow{D}{\left({D}^{{{2}}}+{10}{D}+{25}\right)}={0}\)
\(\displaystyle⟹{D}{\left({D}+{5}\right)}^{{{2}}}={0}\)
\(\displaystyle\Rightarrow{D}{\left({D}+{5}\right)}{\left({D}+{5}\right)}={0}\)
\(\displaystyle\Rightarrow{D}={0},-{5},-{5}.\)
Hence the required general solution is,
\(\displaystyle{y}={c}{1}{e}^{{{0}{x}}}+{\left({c}{2}+{c}{3}{x}\right)}{e}^{{-{5}{x}}}={c}{1}+{\left({c}{2}+{c}{3}{x}\right)}{e}^{{-{5}{x}}},{c}{1},{c}{2},{c}{3}\) are arbitrary constants.
0

Relevant Questions

asked 2021-05-10
Solve the equation:
\(\displaystyle{\left({a}-{x}\right)}{\left.{d}{y}\right.}+{\left({a}+{y}\right)}{\left.{d}{x}\right.}={0}\)
asked 2021-03-22
Solve the equation:
\(\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{2}}+{1}\right)}\)
asked 2021-01-08
Solve the Differential equations \(\displaystyle{\left({D}^{{3}}−{3}{D}+{2}\right)}{y}={0}\)
asked 2021-02-21
Solve the given system of differential equations.
\[Dx+Dy+(D+1)z=0\)
Dx+y=e^{t}\)
Dx+y-2z=50\sin(2t)\)
asked 2021-02-15
Solve
\(\left(d^{2}\frac{y}{dt^{2}}\right)\ +\ 7\left(\frac{dy}{dt}\right)\ +\ 10y=4te^{-3}t\) with
\(y(0)=0,\ y'(0)=\ -1\)
asked 2021-01-02
Solve the Differential equations
\(\displaystyle{\frac{{{\left({d}^{{{2}}}\right)}{y}}}{\rbrace}}{\left\lbrace{d}{\left({t}^{{{2}}}\right\rbrace}+{4}{\left({\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}+{3}{y}={e}^{{-{t}}}\right.}\right.}\)
asked 2021-02-11
Solve the Bernoulli Differential equations. \(\displaystyle{x}{y}′-{y}={x}{y}^{{5}}\)
asked 2021-02-21
Solve the Differential equations \(y'' + 4y = 4 \tan^{2} x\)
asked 2021-02-05
Solve the differential equations
(1) \(\displaystyle{x}{y}'-{2}{y}={x}^{{3}}{e}^{{x}}\)
(2) \(\displaystyle{\left({2}{y}{\left.{d}{x}\right.}+{\left.{d}{y}\right.}\right)}{e}^{{2}}{x}={0}\)
asked 2021-01-02
Solve \(y''\ +\ 3y'\ -\ 10y=x(e^{x}\ +\ 1)\)
...