71.18.16 problem 14
Internal
problem
ID
[14590]
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
:
14
Date
solved
:
Tuesday, January 28, 2025 at 06:43:55 AM
CAS
classification
:
system_of_ODEs
\begin{align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-3 y_{1} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=y_{4} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-5 y_{3} \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.204 (sec). Leaf size: 547
dsolve([diff(y__1(x),x)=0*y__1(x)+1*y__2(x)+0*y__3(x)+0*y__4(x),diff(y__2(x),x)=-3*y__1(x)+0*y__2(x)+2*y__3(x)+0*y__4(x),diff(y__3(x),x)=0*y__1(x)+0*y__2(x)+0*y__3(x)+1*y__4(x),diff(y__4(x),x)=2*y__1(x)+0*y__2(x)-5*y__3(x)+0*y__4(x)],singsol=all)
\begin{align*}
y_{1} \left (x \right ) &= -\frac {c_{1} \left (\sqrt {5}+4\right )^{{3}/{2}} \cos \left (\sqrt {\sqrt {5}+4}\, x \right )}{11}-\frac {c_{2} \left (-\sqrt {5}+4\right )^{{3}/{2}} \cos \left (\sqrt {-\sqrt {5}+4}\, x \right )}{11}-\frac {c_{3} \left (\sqrt {5}+4\right )^{{3}/{2}} \sin \left (\sqrt {\sqrt {5}+4}\, x \right )}{11}-\frac {c_4 \left (-\sqrt {5}+4\right )^{{3}/{2}} \sin \left (\sqrt {-\sqrt {5}+4}\, x \right )}{11}+\frac {8 c_{1} \sqrt {\sqrt {5}+4}\, \cos \left (\sqrt {\sqrt {5}+4}\, x \right )}{11}+\frac {8 c_{2} \sqrt {-\sqrt {5}+4}\, \cos \left (\sqrt {-\sqrt {5}+4}\, x \right )}{11}+\frac {8 c_{3} \sqrt {\sqrt {5}+4}\, \sin \left (\sqrt {\sqrt {5}+4}\, x \right )}{11}+\frac {8 c_4 \sqrt {-\sqrt {5}+4}\, \sin \left (\sqrt {-\sqrt {5}+4}\, x \right )}{11} \\
y_{2} \left (x \right ) &= -c_{1} \sin \left (\sqrt {\sqrt {5}+4}\, x \right )-c_{2} \sin \left (\sqrt {-\sqrt {5}+4}\, x \right )+c_{3} \cos \left (\sqrt {\sqrt {5}+4}\, x \right )+c_4 \cos \left (\sqrt {-\sqrt {5}+4}\, x \right ) \\
y_{3} \left (x \right ) &= \frac {13 c_{1} \sqrt {\sqrt {5}+4}\, \cos \left (\sqrt {\sqrt {5}+4}\, x \right )}{22}+\frac {13 c_{2} \sqrt {-\sqrt {5}+4}\, \cos \left (\sqrt {-\sqrt {5}+4}\, x \right )}{22}+\frac {13 c_{3} \sqrt {\sqrt {5}+4}\, \sin \left (\sqrt {\sqrt {5}+4}\, x \right )}{22}+\frac {13 c_4 \sqrt {-\sqrt {5}+4}\, \sin \left (\sqrt {-\sqrt {5}+4}\, x \right )}{22}-\frac {3 c_{1} \left (\sqrt {5}+4\right )^{{3}/{2}} \cos \left (\sqrt {\sqrt {5}+4}\, x \right )}{22}-\frac {3 c_{2} \left (-\sqrt {5}+4\right )^{{3}/{2}} \cos \left (\sqrt {-\sqrt {5}+4}\, x \right )}{22}-\frac {3 c_{3} \left (\sqrt {5}+4\right )^{{3}/{2}} \sin \left (\sqrt {\sqrt {5}+4}\, x \right )}{22}-\frac {3 c_4 \left (-\sqrt {5}+4\right )^{{3}/{2}} \sin \left (\sqrt {-\sqrt {5}+4}\, x \right )}{22} \\
y_{4} \left (x \right ) &= \frac {c_{1} \sin \left (\sqrt {\sqrt {5}+4}\, x \right ) \sqrt {5}}{2}-\frac {c_{2} \sin \left (\sqrt {-\sqrt {5}+4}\, x \right ) \sqrt {5}}{2}-\frac {c_{3} \cos \left (\sqrt {\sqrt {5}+4}\, x \right ) \sqrt {5}}{2}+\frac {c_4 \cos \left (\sqrt {-\sqrt {5}+4}\, x \right ) \sqrt {5}}{2}+\frac {c_{1} \sin \left (\sqrt {\sqrt {5}+4}\, x \right )}{2}+\frac {c_{2} \sin \left (\sqrt {-\sqrt {5}+4}\, x \right )}{2}-\frac {c_{3} \cos \left (\sqrt {\sqrt {5}+4}\, x \right )}{2}-\frac {c_4 \cos \left (\sqrt {-\sqrt {5}+4}\, x \right )}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 730
DSolve[{D[ y1[x],x]==0*y1[x]+1*y2[x]+0*y3[x]+0*y4[x],D[ y2[x],x]==-3*y1[x]+0*y2[x]+2*y3[x]+0*y4[x],D[ y3[x],x]==0*y1[x]+0*y2[x]+0*y3[x]+1*y4[x],D[ y4[x],x]==2*y1[x]+0*y2[x]-5*y3[x]+0*y4[x]},{y1[x],y2[x],y3[x],y4[x]},x,IncludeSingularSolutions -> True]
\begin{align*}
\text {y1}(x)\to \frac {1}{2} c_3 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{4} c_1 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} x}+5 e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{2} c_4 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {e^{\text {$\#$1} x}}{\text {$\#$1}^3+4 \text {$\#$1}}\&\right ]+\frac {1}{4} c_2 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} x}+5 e^{\text {$\#$1} x}}{\text {$\#$1}^3+4 \text {$\#$1}}\&\right ] \\
\text {y2}(x)\to \frac {1}{2} c_4 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{2} c_3 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1} e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{4} c_2 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} x}+5 e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]-\frac {1}{4} c_1 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {3 \text {$\#$1}^2 e^{\text {$\#$1} x}+11 e^{\text {$\#$1} x}}{\text {$\#$1}^3+4 \text {$\#$1}}\&\right ] \\
\text {y3}(x)\to \frac {1}{2} c_1 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{4} c_3 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} x}+3 e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{2} c_2 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {e^{\text {$\#$1} x}}{\text {$\#$1}^3+4 \text {$\#$1}}\&\right ]+\frac {1}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} x}+3 e^{\text {$\#$1} x}}{\text {$\#$1}^3+4 \text {$\#$1}}\&\right ] \\
\text {y4}(x)\to \frac {1}{2} c_2 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{2} c_1 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1} e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]+\frac {1}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} x}+3 e^{\text {$\#$1} x}}{\text {$\#$1}^2+4}\&\right ]-\frac {1}{4} c_3 \text {RootSum}\left [\text {$\#$1}^4+8 \text {$\#$1}^2+11\&,\frac {5 \text {$\#$1}^2 e^{\text {$\#$1} x}+11 e^{\text {$\#$1} x}}{\text {$\#$1}^3+4 \text {$\#$1}}\&\right ] \\
\end{align*}