23.4.11 problem 11(b)

Internal problem ID [4176]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 11(b)
Date solved : Monday, January 27, 2025 at 08:41:20 AM
CAS classification : system_of_ODEs

\begin{align*} 2 y_{1}^{\prime }\left (x \right )+y_{2}^{\prime }\left (x \right )-4 y_{1} \left (x \right )-y_{2} \left (x \right )&={\mathrm e}^{x}\\ y_{1}^{\prime }\left (x \right )+3 y_{1} \left (x \right )+y_{2} \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 45

dsolve([2*diff(y__1(x),x)+diff(y__2(x),x)-4*y__1(x)-y__2(x)=exp(x),diff(y__1(x),x)+3*y__1(x)+y__2(x)=0],singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= -\frac {3 c_{2} \sin \left (x \right )}{10}-\frac {3 \cos \left (x \right ) c_{1}}{10}-\frac {{\mathrm e}^{x}}{2}+\frac {c_{2} \cos \left (x \right )}{10}-\frac {c_{1} \sin \left (x \right )}{10} \\ y_{2} \left (x \right ) &= c_{2} \sin \left (x \right )+\cos \left (x \right ) c_{1} +2 \,{\mathrm e}^{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 55

DSolve[{2*D[y1[x],x]+D[y2[x],x]-4*y1[x]-y2[x]==Exp[x],D[y1[x],x]+3*y1[x]+y2[x]==0},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to -\frac {e^x}{2}+c_1 \cos (x)-(3 c_1+c_2) \sin (x) \\ \text {y2}(x)\to 2 e^x+c_2 \cos (x)+(10 c_1+3 c_2) \sin (x) \\ \end{align*}