71.18.2 problem 2
Internal
problem
ID
[14576]
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
:
2
Date
solved
:
Tuesday, January 28, 2025 at 06:43:42 AM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )-2 y_{1} \left (x \right )+\sin \left (2 x \right )\\ y_{2}^{\prime }\left (x \right )&=-3 y_{1} \left (x \right )+y_{2} \left (x \right )-2 \cos \left (3 x \right ) \end{align*}
✓ Solution by Maple
Time used: 5.583 (sec). Leaf size: 145
dsolve([diff(y__1(x),x)=y__2(x)-2*y__1(x)+sin(2*x),diff(y__2(x),x)=-3*y__1(x)+y__2(x)-2*cos(3*x)],singsol=all)
\begin{align*}
y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} +\frac {7 \sin \left (2 x \right )}{13}-\frac {4 \cos \left (2 x \right )}{13}+\frac {16 \cos \left (3 x \right )}{73}-\frac {6 \sin \left (3 x \right )}{73} \\
y_{2} \left (x \right ) &= \frac {3 c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1}}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1}}{2}+\frac {6 \cos \left (2 x \right )}{13}+\frac {9 \sin \left (2 x \right )}{13}-\frac {60 \sin \left (3 x \right )}{73}+\frac {14 \cos \left (3 x \right )}{73} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.016 (sec). Leaf size: 539
DSolve[{D[ y1[x],x]==y2[x]-2*y1[x]+Sin[2*x],D[ y2[x],x]==-3*y1[x]+y2[x]-2*Cos[3*x]},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
\begin{align*}
\text {y1}(x)\to \frac {1}{3} e^{-x/2} \left (3 \left (\cos \left (\frac {\sqrt {3} x}{2}\right )-\sqrt {3} \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \int _1^x\frac {1}{3} e^{\frac {K[1]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) \sin (2 K[1])+\sqrt {3} (4 \cos (3 K[1])+3 \sin (2 K[1])) \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )dK[1]+2 \sqrt {3} \sin \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x2 e^{\frac {K[2]}{2}} \left (\sqrt {3} \sin (2 K[2]) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )+\cos (3 K[2]) \left (\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )-\cos \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )\right )dK[2]+3 c_1 \cos \left (\frac {\sqrt {3} x}{2}\right )-3 \sqrt {3} c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )+2 \sqrt {3} c_2 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \\
\text {y2}(x)\to e^{-x/2} \left (-2 \sqrt {3} \sin \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x\frac {1}{3} e^{\frac {K[1]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) \sin (2 K[1])+\sqrt {3} (4 \cos (3 K[1])+3 \sin (2 K[1])) \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )dK[1]+\left (\sqrt {3} \sin \left (\frac {\sqrt {3} x}{2}\right )+\cos \left (\frac {\sqrt {3} x}{2}\right )\right ) \int _1^x2 e^{\frac {K[2]}{2}} \left (\sqrt {3} \sin (2 K[2]) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )+\cos (3 K[2]) \left (\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )-\cos \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )\right )dK[2]+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )-2 \sqrt {3} c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )+\sqrt {3} c_2 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \\
\end{align*}