18.16 problem 14

Internal problem ID [12854]

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.
ODE order: 1.
ODE degree: 1.

Solve \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: 0.109 (sec). Leaf size: 548

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 (4+\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}-\frac {c_{2} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{11}-\frac {c_{3} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}-\frac {c_{4} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{11}+\frac {8 c_{1} \sqrt {4+\sqrt {5}}\, \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}+\frac {8 c_{2} \sqrt {4-\sqrt {5}}\, \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{11}+\frac {8 c_{3} \sqrt {4+\sqrt {5}}\, \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}+\frac {8 c_{4} \sqrt {4-\sqrt {5}}\, \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{11} \\ y_{2} \left (x \right ) &= -c_{1} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )-c_{2} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )+c_{3} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )+c_{4} \cos \left (\sqrt {4-\sqrt {5}}\, x \right ) \\ y_{3} \left (x \right ) &= \frac {13 c_{1} \sqrt {4+\sqrt {5}}\, \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}+\frac {13 c_{2} \sqrt {4-\sqrt {5}}\, \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{22}+\frac {13 c_{3} \sqrt {4+\sqrt {5}}\, \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}+\frac {13 c_{4} \sqrt {4-\sqrt {5}}\, \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{22}-\frac {3 c_{1} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}-\frac {3 c_{2} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{22}-\frac {3 c_{3} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}-\frac {3 c_{4} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{22} \\ y_{4} \left (x \right ) &= \frac {c_{1} \sin \left (\sqrt {4+\sqrt {5}}\, x \right ) \sqrt {5}}{2}-\frac {c_{2} \sin \left (\sqrt {4-\sqrt {5}}\, x \right ) \sqrt {5}}{2}-\frac {c_{3} \cos \left (\sqrt {4+\sqrt {5}}\, x \right ) \sqrt {5}}{2}+\frac {c_{4} \cos \left (\sqrt {4-\sqrt {5}}\, x \right ) \sqrt {5}}{2}+\frac {c_{1} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{2}+\frac {c_{2} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{2}-\frac {c_{3} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{2}-\frac {c_{4} \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 730

DSolve[{y1'[x]==0*y1[x]+1*y2[x]+0*y3[x]+0*y4[x],y2'[x]==-3*y1[x]+0*y2[x]+2*y3[x]+0*y4[x],y3'[x]==0*y1[x]+0*y2[x]+0*y3[x]+1*y4[x],y4'[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*}