49.24.3 problem 5

Internal problem ID [7771]
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 : 5
Date solved : Monday, January 27, 2025 at 03:21:38 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 35

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

\begin{align*} y_{1} \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{3 x}}{3}+\frac {1}{6} \\ y_{2} \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{2}+\frac {2 \,{\mathrm e}^{3 x}}{3}-\frac {1}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.086 (sec). Leaf size: 46

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

\begin{align*} \text {y1}(x)\to \frac {1}{6} \left (e^x-1\right )^2 \left (2 e^x+1\right ) \\ \text {y2}(x)\to \frac {1}{6} \left (-3 e^{2 x}+4 e^{3 x}-1\right ) \\ \end{align*}