71.18.3 problem 3
Internal
problem
ID
[14577]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
8.
Linear
Systems
of
First-Order
Differential
Equations.
Exercises
8.3
page
379
Problem
number
:
3
Date
solved
:
Tuesday, January 28, 2025 at 06:43:45 AM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (x \right )&=2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=2 y_{3} \left (x \right )-y_{1} \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.190 (sec). Leaf size: 106
dsolve([diff(y__1(x),x)=2*y__2(x),diff(y__2(x),x)=3*y__1(x),diff(y__3(x),x)=2*y__3(x)-y__1(x)],singsol=all)
\begin{align*}
y_{1} \left (x \right ) &= {\mathrm e}^{\sqrt {6}\, x} c_{2} +c_{3} {\mathrm e}^{-\sqrt {6}\, x} \\
y_{2} \left (x \right ) &= \frac {\sqrt {6}\, \left ({\mathrm e}^{\sqrt {6}\, x} c_{2} -c_{3} {\mathrm e}^{-\sqrt {6}\, x}\right )}{2} \\
y_{3} \left (x \right ) &= \frac {2 \,{\mathrm e}^{2 x} c_{1}}{\left (2+\sqrt {6}\right ) \left (-2+\sqrt {6}\right )}+\frac {{\mathrm e}^{-\sqrt {6}\, x} c_{3}}{2+\sqrt {6}}-\frac {{\mathrm e}^{\sqrt {6}\, x} c_{2}}{-2+\sqrt {6}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.020 (sec). Leaf size: 232
DSolve[{D[ y1[x],x]==2*y2[x],D[ y2[x],x]==3*y1[x],D[ y3[x],x]==2*y3[x]-y1[x]},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions -> True]
\begin{align*}
\text {y1}(x)\to \frac {1}{6} e^{-\sqrt {6} x} \left (3 c_1 \left (e^{2 \sqrt {6} x}+1\right )+\sqrt {6} c_2 \left (e^{2 \sqrt {6} x}-1\right )\right ) \\
\text {y2}(x)\to \frac {1}{4} e^{-\sqrt {6} x} \left (\sqrt {6} c_1 \left (e^{2 \sqrt {6} x}-1\right )+2 c_2 \left (e^{2 \sqrt {6} x}+1\right )\right ) \\
\text {y3}(x)\to \frac {1}{12} e^{-\sqrt {6} x} \left (2 \left (c_2 \left (-\left (3+\sqrt {6}\right ) e^{2 \sqrt {6} x}+6 e^{\left (2+\sqrt {6}\right ) x}-3+\sqrt {6}\right )+6 c_3 e^{\left (2+\sqrt {6}\right ) x}\right )-3 c_1 \left (\left (2+\sqrt {6}\right ) e^{2 \sqrt {6} x}-4 e^{\left (2+\sqrt {6}\right ) x}+2-\sqrt {6}\right )\right ) \\
\end{align*}