71.18.17 problem 15

Internal problem ID [14591]
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 : 15
Date solved : Tuesday, January 28, 2025 at 06:43:59 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )+2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-2 y_{1} \left (x \right )+3 y_{2} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=y_{3} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=2 y_{4} \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 61

dsolve([diff(y__1(x),x)=3*y__1(x)+2*y__2(x)+0*y__3(x)+0*y__4(x),diff(y__2(x),x)=-2*y__1(x)+3*y__2(x)+0*y__3(x)+0*y__4(x),diff(y__3(x),x)=0*y__1(x)+0*y__2(x)+1*y__3(x)+0*y__4(x),diff(y__4(x),x)=0*y__1(x)+0*y__2(x)-0*y__3(x)+2*y__4(x)],singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{3 x} \left (c_{1} \sin \left (2 x \right )+c_{2} \cos \left (2 x \right )\right ) \\ y_{2} \left (x \right ) &= -{\mathrm e}^{3 x} \left (\sin \left (2 x \right ) c_{2} -\cos \left (2 x \right ) c_{1} \right ) \\ y_{3} \left (x \right ) &= c_4 \,{\mathrm e}^{x} \\ y_{4} \left (x \right ) &= c_{3} {\mathrm e}^{2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 255

DSolve[{D[ y1[x],x]==3*y1[x]+2*y2[x]+0*y3[x]+0*y4[x],D[ y2[x],x]==-2*y1[x]+3*y2[x]+0*y3[x]+0*y4[x],D[ y3[x],x]==0*y1[x]+0*y2[x]+1*y3[x]+0*y4[x],D[ y4[x],x]==0*y1[x]+0*y2[x]-0*y3[x]+2*y4[x]},{y1[x],y2[x],y3[x],y4[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to e^{3 x} (c_1 \cos (2 x)+c_2 \sin (2 x)) \\ \text {y2}(x)\to e^{3 x} (c_2 \cos (2 x)-c_1 \sin (2 x)) \\ \text {y3}(x)\to c_3 e^x \\ \text {y4}(x)\to c_4 e^{2 x} \\ \text {y1}(x)\to e^{3 x} (c_1 \cos (2 x)+c_2 \sin (2 x)) \\ \text {y2}(x)\to e^{3 x} (c_2 \cos (2 x)-c_1 \sin (2 x)) \\ \text {y3}(x)\to c_3 e^x \\ \text {y4}(x)\to 0 \\ \text {y1}(x)\to e^{3 x} (c_1 \cos (2 x)+c_2 \sin (2 x)) \\ \text {y2}(x)\to e^{3 x} (c_2 \cos (2 x)-c_1 \sin (2 x)) \\ \text {y3}(x)\to 0 \\ \text {y4}(x)\to c_4 e^{2 x} \\ \text {y1}(x)\to e^{3 x} (c_1 \cos (2 x)+c_2 \sin (2 x)) \\ \text {y2}(x)\to e^{3 x} (c_2 \cos (2 x)-c_1 \sin (2 x)) \\ \text {y3}(x)\to 0 \\ \text {y4}(x)\to 0 \\ \end{align*}