Internal problem ID [12853]
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: 13.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )+2 y_{2} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=3 y_{3} \left (x \right )-4 y_{4} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=4 y_{3} \left (x \right )+3 y_{4} \left (x \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 82
dsolve([diff(y__1(x),x)=2*y__1(x)+1*y__2(x)+0*y__3(x)+0*y__4(x),diff(y__2(x),x)=-1*y__1(x)+2*y__2(x)+0*y__3(x)+0*y__4(x),diff(y__3(x),x)=0*y__1(x)+0*y__2(x)+3*y__3(x)-4*y__4(x),diff(y__4(x),x)=0*y__1(x)+0*y__2(x)+4*y__3(x)+3*y__4(x)],singsol=all)
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{3} +c_{4} \cos \left (x \right )\right ) \\ y_{2} \left (x \right ) &= -{\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{4} -\cos \left (x \right ) c_{3} \right ) \\ y_{3} \left (x \right ) &= {\mathrm e}^{3 x} \left (\cos \left (4 x \right ) c_{2} +\sin \left (4 x \right ) c_{1} \right ) \\ y_{4} \left (x \right ) &= -{\mathrm e}^{3 x} \left (\cos \left (4 x \right ) c_{1} -\sin \left (4 x \right ) c_{2} \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 92
DSolve[{y1'[x]==2*y1[x]+1*y2[x]+0*y3[x]+0*y4[x],y2'[x]==-1*y1[x]+2*y2[x]+0*y3[x]+0*y4[x],y3'[x]==0*y1[x]+0*y2[x]+3*y3[x]-4*y4[x],y4'[x]==0*y1[x]+0*y2[x]+4*y3[x]+3*y4[x]},{y1[x],y2[x],y3[x],y4[x]},x,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(x)\to e^{2 x} (c_1 \cos (x)+c_2 \sin (x)) \\ \text {y2}(x)\to e^{2 x} (c_2 \cos (x)-c_1 \sin (x)) \\ \text {y3}(x)\to e^{3 x} (c_3 \cos (4 x)-c_4 \sin (4 x)) \\ \text {y4}(x)\to e^{3 x} (c_4 \cos (4 x)+c_3 \sin (4 x)) \\ \end{align*}