82.56.4 problem Ex. 15

Internal problem ID [19044]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter XI. Ordinary differential equations with more than two variables. End of chapter problems at page 143
Problem number : Ex. 15
Date solved : Tuesday, January 28, 2025 at 12:46:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=n y \left (t \right )-m z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=L z \left (t \right )-m x \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=m x \left (t \right )-L y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.619 (sec). Leaf size: 10311

dsolve([diff(x(t),t)=n*y(t)-m*z(t),diff(y(t),t)=L*z(t)-m*x(t),diff(z(t),t)=m*x(t)-L*y(t)],singsol=all)
 

\begin{align*} x &= c_{1} {\mathrm e}^{-\frac {\left (12 i L^{2} \sqrt {3}+i \left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{2}/{3}} \sqrt {3}+12 i \sqrt {3}\, m^{2}+12 i \sqrt {3}\, m n -12 L^{2}+\left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{2}/{3}}-12 m^{2}-12 m n \right ) t}{12 \left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{1}/{3}}}}+c_{2} {\mathrm e}^{\frac {\left (12 i L^{2} \sqrt {3}+i \left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{2}/{3}} \sqrt {3}+12 i \sqrt {3}\, m^{2}+12 i \sqrt {3}\, m n +12 L^{2}-\left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{2}/{3}}+12 m^{2}+12 m n \right ) t}{12 \left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{1}/{3}}}}+c_3 \,{\mathrm e}^{-\frac {\left (12 L^{2}-\left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{2}/{3}}+12 m^{2}+12 m n \right ) t}{6 \left (-108 L \,m^{2}+108 L m n +12 \sqrt {12 L^{6}+36 L^{4} m^{2}+36 L^{4} m n +117 L^{2} m^{4}-90 L^{2} m^{3} n +117 L^{2} m^{2} n^{2}+12 m^{6}+36 m^{5} n +36 m^{4} n^{2}+12 m^{3} n^{3}}\right )^{{1}/{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 715

DSolve[{D[x[t],t]==n*y[t]-m*z[t],D[y[t],t]==L*z[t]-m*x[t],D[z[t],t]==m*x[t]-L*y[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {L n e^{\text {$\#$1} t}-\text {$\#$1} m e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {L m e^{\text {$\#$1} t}+\text {$\#$1} n e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+L^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ] \\ y(t)\to c_1 (-m) \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-L e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1} L e^{\text {$\#$1} t}+m^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+m^2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ] \\ z(t)\to c_1 m \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {L e^{\text {$\#$1} t}+\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {m n e^{\text {$\#$1} t}-\text {$\#$1} L e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1} L^2+\text {$\#$1} m^2+\text {$\#$1} m n+L m^2-L m n\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+m n e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+L^2+m^2+m n}\&\right ] \\ \end{align*}