Internal problem ID [5350]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\relax (x )&=y_{1} \relax (x )+y_{2} \relax (x )\\ y_{2}^{\prime }\relax (x )&=y_{1} \relax (x )+y_{2} \relax (x )+{\mathrm e}^{3 x} \end {align*}
With initial conditions \[ [y_{1} \relax (0) = 0, y_{2} \relax (0) = 0] \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 36
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],[y__1(x), y__2(x)], singsol=all)
\[ y_{1} \relax (x ) = -\frac {{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{3 x}}{3}+\frac {1}{6} \] \[ y_{2} \relax (x ) = -\frac {{\mathrm e}^{2 x}}{2}+\frac {2 \,{\mathrm e}^{3 x}}{3}-\frac {1}{6} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 45
DSolve[{y1'[x]==y1[x]+y2[x],y2'[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 (e^{2 x} \left (4 e^x-3\right )-1\right ) \\ \end{align*}