Internal problem ID [12840]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )-2 y_{1} \left (x \right )+2 \cos \left (x \right ) \sin \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-3 y_{1} \left (x \right )+y_{2} \left (x \right )-8 \cos \left (x \right )^{3}+6 \cos \left (x \right ) \end {align*}
✓ Solution by Maple
Time used: 1.578 (sec). Leaf size: 146
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 {16 \cos \left (3 x \right )}{73}-\frac {4 \cos \left (2 x \right )}{13}-\frac {6 \sin \left (3 x \right )}{73}+\frac {7 \sin \left (2 x \right )}{13} \\ 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 {60 \sin \left (3 x \right )}{73}+\frac {9 \sin \left (2 x \right )}{13}+\frac {14 \cos \left (3 x \right )}{73}+\frac {6 \cos \left (2 x \right )}{13} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.455 (sec). Leaf size: 223
DSolve[{y1'[x]==y2[x]-2*y1[x]+Sin[2*x],y2'[x]==-3*y1[x]+y2[x]-2*Cos[3*x]},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(x)\to \frac {7}{13} \sin (2 x)-\frac {6}{73} \sin (3 x)-\frac {4}{13} \cos (2 x)+\frac {16}{73} \cos (3 x)+c_1 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )-\sqrt {3} c_1 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+\frac {2 c_2 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )}{\sqrt {3}} \\ \text {y2}(x)\to \frac {9}{13} \sin (2 x)-\frac {60}{73} \sin (3 x)+\frac {6}{13} \cos (2 x)+\frac {14}{73} \cos (3 x)+c_2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )-2 \sqrt {3} c_1 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+\sqrt {3} c_2 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right ) \\ \end{align*}