71.18.6 problem 5 c
Internal
problem
ID
[14580]
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
:
5
c
Date
solved
:
Tuesday, January 28, 2025 at 06:43:47 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 )+4 x -2\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right )+3 x \end{align*}
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 35
dsolve([diff(y__1(x),x)=2*y__1(x)-3*y__2(x)+4*x-2,diff(y__2(x),x)=y__1(x)-2*y__2(x)+3*x],singsol=all)
\begin{align*}
y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1} +x \\
y_{2} \left (x \right ) &= c_{2} {\mathrm e}^{-x}+\frac {{\mathrm e}^{x} c_{1}}{3}-1+2 x \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.445 (sec). Leaf size: 333
DSolve[{D[ y1[x],x]==-2*y1[x]-3*y2[x]+4*x-2,D[ y2[x],x]==y1[x]-2*y2[x]+3*x},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
\begin{align*}
\text {y1}(x)\to e^{-2 x} \left (\cos \left (\sqrt {3} x\right ) \int _1^xe^{2 K[1]} \left (\cos \left (\sqrt {3} K[1]\right ) (4 K[1]-2)+3 \sqrt {3} K[1] \sin \left (\sqrt {3} K[1]\right )\right )dK[1]-\sqrt {3} \sin \left (\sqrt {3} x\right ) \int _1^x\frac {1}{3} e^{2 K[2]} \left (9 \cos \left (\sqrt {3} K[2]\right ) K[2]+2 \sqrt {3} (1-2 K[2]) \sin \left (\sqrt {3} K[2]\right )\right )dK[2]+c_1 \cos \left (\sqrt {3} x\right )-\sqrt {3} c_2 \sin \left (\sqrt {3} x\right )\right ) \\
\text {y2}(x)\to \frac {1}{3} e^{-2 x} \left (3 \cos \left (\sqrt {3} x\right ) \int _1^x\frac {1}{3} e^{2 K[2]} \left (9 \cos \left (\sqrt {3} K[2]\right ) K[2]+2 \sqrt {3} (1-2 K[2]) \sin \left (\sqrt {3} K[2]\right )\right )dK[2]+\sqrt {3} \sin \left (\sqrt {3} x\right ) \int _1^xe^{2 K[1]} \left (\cos \left (\sqrt {3} K[1]\right ) (4 K[1]-2)+3 \sqrt {3} K[1] \sin \left (\sqrt {3} K[1]\right )\right )dK[1]+3 c_2 \cos \left (\sqrt {3} x\right )+\sqrt {3} c_1 \sin \left (\sqrt {3} x\right )\right ) \\
\end{align*}