71.18.1 problem 1
Internal
problem
ID
[14575]
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
:
1
Date
solved
:
Tuesday, January 28, 2025 at 06:43:40 AM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )+5 \,{\mathrm e}^{x}\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+4 y_{2} \left (x \right )-2 \,{\mathrm e}^{-x} \end{align*}
✓ Solution by Maple
Time used: 0.811 (sec). Leaf size: 111
dsolve([diff(y__1(x),x)=2*y__1(x)-3*y__2(x)+5*exp(x),diff(y__2(x),x)=y__1(x)+4*y__2(x)-2*exp(-x)],singsol=all)
\begin{align*}
y_{1} \left (x \right ) &= {\mathrm e}^{3 x} \cos \left (\sqrt {2}\, x \right ) c_{2} +{\mathrm e}^{3 x} \sin \left (\sqrt {2}\, x \right ) c_{1} +\frac {{\mathrm e}^{-x}}{3}-\frac {5 \,{\mathrm e}^{x}}{2} \\
y_{2} \left (x \right ) &= -\frac {{\mathrm e}^{3 x} \cos \left (\sqrt {2}\, x \right ) c_{2}}{3}+\frac {{\mathrm e}^{3 x} \sqrt {2}\, \sin \left (\sqrt {2}\, x \right ) c_{2}}{3}-\frac {{\mathrm e}^{3 x} \sin \left (\sqrt {2}\, x \right ) c_{1}}{3}-\frac {{\mathrm e}^{3 x} \sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) c_{1}}{3}+\frac {{\mathrm e}^{-x}}{3}+\frac {5 \,{\mathrm e}^{x}}{6} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.323 (sec). Leaf size: 420
DSolve[{D[ y1[x],x]==2*y1[x]-3*y2[x]+5*Exp[x],D[ y2[x],x]==y1[x]+4*y2[x]-2*Exp[-x]},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
\begin{align*}
\text {y1}(x)\to -\frac {1}{2} e^{3 x} \left (\left (\sqrt {2} \sin \left (\sqrt {2} x\right )-2 \cos \left (\sqrt {2} x\right )\right ) \int _1^x\left (5 e^{-2 K[1]} \cos \left (\sqrt {2} K[1]\right )+\frac {e^{-4 K[1]} \left (-6+5 e^{2 K[1]}\right ) \sin \left (\sqrt {2} K[1]\right )}{\sqrt {2}}\right )dK[1]+3 \sqrt {2} \sin \left (\sqrt {2} x\right ) \int _1^x\frac {1}{2} e^{-4 K[2]} \left (-4 \cos \left (\sqrt {2} K[2]\right )-\sqrt {2} \left (-2+5 e^{2 K[2]}\right ) \sin \left (\sqrt {2} K[2]\right )\right )dK[2]-2 c_1 \cos \left (\sqrt {2} x\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} x\right )+3 \sqrt {2} c_2 \sin \left (\sqrt {2} x\right )\right ) \\
\text {y2}(x)\to \frac {1}{2} e^{3 x} \left (\sqrt {2} \sin \left (\sqrt {2} x\right ) \int _1^x\left (5 e^{-2 K[1]} \cos \left (\sqrt {2} K[1]\right )+\frac {e^{-4 K[1]} \left (-6+5 e^{2 K[1]}\right ) \sin \left (\sqrt {2} K[1]\right )}{\sqrt {2}}\right )dK[1]+\left (\sqrt {2} \sin \left (\sqrt {2} x\right )+2 \cos \left (\sqrt {2} x\right )\right ) \int _1^x\frac {1}{2} e^{-4 K[2]} \left (-4 \cos \left (\sqrt {2} K[2]\right )-\sqrt {2} \left (-2+5 e^{2 K[2]}\right ) \sin \left (\sqrt {2} K[2]\right )\right )dK[2]+2 c_2 \cos \left (\sqrt {2} x\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} x\right )+\sqrt {2} c_2 \sin \left (\sqrt {2} x\right )\right ) \\
\end{align*}