49.24.2 problem 4

Internal problem ID [7770]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page 250
Problem number : 4
Date solved : Monday, January 27, 2025 at 03:21:37 PM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=6 y_{1} \left (x \right )+y_{2} \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 33

dsolve([diff(y__1(x),x) = y__2(x), diff(y__2(x),x) = 6*y__1(x)+y__2(x), y__1(0) = 1, y__2(0) = -1], singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= \frac {4 \,{\mathrm e}^{-2 x}}{5}+\frac {{\mathrm e}^{3 x}}{5} \\ y_{2} \left (x \right ) &= -\frac {8 \,{\mathrm e}^{-2 x}}{5}+\frac {3 \,{\mathrm e}^{3 x}}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 42

DSolve[{D[ y1[x],x]==y2[x],D[ y2[x],x]==6*y1[x]+y2[x]},{y1[0]==1,y2[0]==-1},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{5} e^{-2 x} \left (e^{5 x}+4\right ) \\ \text {y2}(x)\to \frac {1}{5} e^{-2 x} \left (3 e^{5 x}-8\right ) \\ \end{align*}