60.9.10 problem 1865

Internal problem ID [11864]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1865
Date solved : Monday, January 27, 2025 at 11:43:57 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}\\ \frac {d}{d t}y \left (t \right )&=a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2} \end{align*}

✓ Solution by Maple

Time used: 0.173 (sec). Leaf size: 333

dsolve({diff(x(t),t)=a__1*x(t)+b__1*y(t)+c__1,diff(y(t),t)=a__2*x(t)+b__2*y(t)+c__2},singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (\frac {a_{1}}{2}+\frac {b_{2}}{2}+\frac {\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}{2}\right ) t} c_4 +{\mathrm e}^{\left (\frac {a_{1}}{2}+\frac {b_{2}}{2}-\frac {\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}{2}\right ) t} c_3 +\frac {b_{1} c_{2} -b_{2} c_{1}}{a_{1} b_{2} -a_{2} b_{1}} \\ y \left (t \right ) &= \frac {-\frac {a_{1} \left ({\mathrm e}^{\frac {\left (a_{1} +b_{2} +\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_4 \left (a_{1} b_{2} -a_{2} b_{1} \right )+{\mathrm e}^{\frac {\left (a_{1} +b_{2} -\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_3 \left (a_{1} b_{2} -a_{2} b_{1} \right )-b_{2} c_{1} +b_{1} c_{2} \right ) \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{a_{1} b_{2} -a_{2} b_{1}}+\frac {\left (a_{1} +b_{2} +\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) {\mathrm e}^{\frac {\left (a_{1} +b_{2} +\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_4 \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{2}+\frac {\left (a_{1} +b_{2} -\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) {\mathrm e}^{\frac {\left (a_{1} +b_{2} -\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\right ) t}{2}} c_3 \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{2}-c_{1} \left (2 a_{1} b_{2} -2 a_{2} b_{1} \right )}{\left (2 a_{1} b_{2} -2 a_{2} b_{1} \right ) b_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.098 (sec). Leaf size: 1684

DSolve[{D[x[t],t]==a1*x[t]+b1*y[t]+c1,D[y[t],t]==a2*x[t]+b2*y[t]+c2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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