60.9.54 problem 1909

Internal problem ID [11908]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1909
Date solved : Monday, January 27, 2025 at 11:44:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=a x \left (t \right )+g y \left (t \right )+\beta z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=g x \left (t \right )+b y \left (t \right )+\alpha z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=\beta x \left (t \right )+\alpha y \left (t \right )+c z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 19.314 (sec). Leaf size: 32424

dsolve([diff(x(t),t)=a*x(t)+g*y(t)+beta*z(t),diff(y(t),t)=g*x(t)+b*y(t)+alpha*z(t),diff(z(t),t)=beta*x(t)+alpha*y(t)+c*z(t)],singsol=all)
 

\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 1639

DSolve[{D[x[t],t]==a*x[t]+g*y[t]+\[Beta]*z[t],D[y[t],t]==g*x[t]+b*y[t]+\[Alpha]*z[t],D[z[t],t]==\[Beta]*x[t]+\[Alpha]*y[t]+c*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -c_3 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {-b \beta e^{\text {$\#$1} t}+\alpha g e^{\text {$\#$1} t}+\text {$\#$1} \beta e^{\text {$\#$1} t}}{-3 \text {$\#$1}^2+2 \text {$\#$1} a+2 \text {$\#$1} b+2 \text {$\#$1} c+\alpha ^2-a b-a c+\beta ^2-b c+g^2}\&\right ]+c_2 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {-c g e^{\text {$\#$1} t}+\text {$\#$1} g e^{\text {$\#$1} t}+\alpha \beta e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ]+c_1 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+b c e^{\text {$\#$1} t}-\text {$\#$1} b e^{\text {$\#$1} t}-\text {$\#$1} c e^{\text {$\#$1} t}+\alpha ^2 \left (-e^{\text {$\#$1} t}\right )}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ] \\ y(t)\to c_1 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {-c g e^{\text {$\#$1} t}+\text {$\#$1} g e^{\text {$\#$1} t}+\alpha \beta e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ]+c_3 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {-a \alpha e^{\text {$\#$1} t}+\beta g e^{\text {$\#$1} t}+\text {$\#$1} \alpha e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ]+c_2 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+a c e^{\text {$\#$1} t}-\text {$\#$1} a e^{\text {$\#$1} t}-\text {$\#$1} c e^{\text {$\#$1} t}+\beta ^2 \left (-e^{\text {$\#$1} t}\right )}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ] \\ z(t)\to -c_1 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {-b \beta e^{\text {$\#$1} t}+\alpha g e^{\text {$\#$1} t}+\text {$\#$1} \beta e^{\text {$\#$1} t}}{-3 \text {$\#$1}^2+2 \text {$\#$1} a+2 \text {$\#$1} b+2 \text {$\#$1} c+\alpha ^2-a b-a c+\beta ^2-b c+g^2}\&\right ]+c_2 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {-a \alpha e^{\text {$\#$1} t}+\beta g e^{\text {$\#$1} t}+\text {$\#$1} \alpha e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ]+c_3 \text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1}^2 b+\text {$\#$1}^2 c+\text {$\#$1} \alpha ^2-\text {$\#$1} a b-\text {$\#$1} a c+\text {$\#$1} \beta ^2-\text {$\#$1} b c+\text {$\#$1} g^2-a \alpha ^2+a b c-b \beta ^2-c g^2+2 \alpha \beta g\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+a b e^{\text {$\#$1} t}-\text {$\#$1} a e^{\text {$\#$1} t}-\text {$\#$1} b e^{\text {$\#$1} t}+g^2 \left (-e^{\text {$\#$1} t}\right )}{3 \text {$\#$1}^2-2 \text {$\#$1} a-2 \text {$\#$1} b-2 \text {$\#$1} c-\alpha ^2+a b+a c-\beta ^2+b c-g^2}\&\right ] \\ \end{align*}