Internal problem ID [10191]
Internal file name [OUTPUT/9138_Monday_June_06_2022_06_44_22_AM_10595268/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1869.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-2 x \left (t \right )-y \left (t \right )+{\mathrm e}^{2 t}+t\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )-3 y \left (t \right )+{\mathrm e}^{t}-1 \end {align*}
The system is \begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-2 x \left (t \right )-y \left (t \right )+{\mathrm e}^{2 t}+t\tag {1}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )-3 y \left (t \right )+{\mathrm e}^{t}-1\tag {2} \end {align*}
Since the left side is the same, this implies \begin {align*} -2 x \left (t \right )-y \left (t \right )+{\mathrm e}^{2 t}+t&=x \left (t \right )-3 y \left (t \right )+{\mathrm e}^{t}-1\\ y \left (t \right )&=-\frac {{\mathrm e}^{2 t}}{2}+\frac {3 x \left (t \right )}{2}+\frac {{\mathrm e}^{t}}{2}-\frac {t}{2}-\frac {1}{2}\tag {3} \end {align*}
Taking derivative of the above w.r.t. \(t\) gives \begin {align*} y^{\prime }\left (t \right )&=-{\mathrm e}^{2 t}+\frac {3 x^{\prime }\left (t \right )}{2}+\frac {{\mathrm e}^{t}}{2}-\frac {1}{2}\tag {4} \end {align*}
Substituting (3,4) in (1) to eliminate \(y \left (t \right ),y^{\prime }\left (t \right )\) gives \begin {align*} \frac {5 x^{\prime }\left (t \right )}{2}-{\mathrm e}^{2 t}+\frac {{\mathrm e}^{t}}{2}-\frac {1}{2} &= -\frac {7 x \left (t \right )}{2}+\frac {3 \,{\mathrm e}^{2 t}}{2}-\frac {{\mathrm e}^{t}}{2}+\frac {3 t}{2}+\frac {1}{2}\\ x^{\prime }\left (t \right ) &= -\frac {7 x \left (t \right )}{5}+{\mathrm e}^{2 t}-\frac {2 \,{\mathrm e}^{t}}{5}+\frac {3 t}{5}+\frac {2}{5}\tag {5} \end {align*}
Which is now solved for \(x \left (t \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} x^{\prime }\left (t \right ) + p(t)x \left (t \right ) &= q(t) \end {align*}
Where here \begin {align*} p(t) &={\frac {7}{5}}\\ q(t) &={\mathrm e}^{2 t}-\frac {2 \,{\mathrm e}^{t}}{5}+\frac {3 t}{5}+\frac {2}{5} \end {align*}
Hence the ode is \begin {align*} x^{\prime }\left (t \right )+\frac {7 x \left (t \right )}{5} = {\mathrm e}^{2 t}-\frac {2 \,{\mathrm e}^{t}}{5}+\frac {3 t}{5}+\frac {2}{5} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {7}{5}d t} \\ &= {\mathrm e}^{\frac {7 t}{5}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}}\left ( \mu x\right ) &= \left (\mu \right ) \left ({\mathrm e}^{2 t}-\frac {2 \,{\mathrm e}^{t}}{5}+\frac {3 t}{5}+\frac {2}{5}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \left ({\mathrm e}^{\frac {7 t}{5}} x\right ) &= \left ({\mathrm e}^{\frac {7 t}{5}}\right ) \left ({\mathrm e}^{2 t}-\frac {2 \,{\mathrm e}^{t}}{5}+\frac {3 t}{5}+\frac {2}{5}\right )\\ \mathrm {d} \left ({\mathrm e}^{\frac {7 t}{5}} x\right ) &= \left (\frac {\left (5 \,{\mathrm e}^{2 t}-2 \,{\mathrm e}^{t}+3 t +2\right ) {\mathrm e}^{\frac {7 t}{5}}}{5}\right )\, \mathrm {d} t \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{\frac {7 t}{5}} x &= \int {\frac {\left (5 \,{\mathrm e}^{2 t}-2 \,{\mathrm e}^{t}+3 t +2\right ) {\mathrm e}^{\frac {7 t}{5}}}{5}\,\mathrm {d} t}\\ {\mathrm e}^{\frac {7 t}{5}} x &= \frac {3 \,{\mathrm e}^{\frac {7 t}{5}} t}{7}-\frac {{\mathrm e}^{\frac {7 t}{5}}}{49}-\frac {{\mathrm e}^{\frac {12 t}{5}}}{6}+\frac {5 \,{\mathrm e}^{\frac {17 t}{5}}}{17} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\frac {7 t}{5}}\) results in \begin {align*} x \left (t \right ) &= {\mathrm e}^{-\frac {7 t}{5}} \left (\frac {3 \,{\mathrm e}^{\frac {7 t}{5}} t}{7}-\frac {{\mathrm e}^{\frac {7 t}{5}}}{49}-\frac {{\mathrm e}^{\frac {12 t}{5}}}{6}+\frac {5 \,{\mathrm e}^{\frac {17 t}{5}}}{17}\right )+c_{1} {\mathrm e}^{-\frac {7 t}{5}} \end {align*}
which simplifies to \begin {align*} x \left (t \right ) &= -\frac {\left (-1470 \,{\mathrm e}^{\frac {17 t}{5}}+833 \,{\mathrm e}^{\frac {12 t}{5}}-2142 \,{\mathrm e}^{\frac {7 t}{5}} t +102 \,{\mathrm e}^{\frac {7 t}{5}}-4998 c_{1} \right ) {\mathrm e}^{-\frac {7 t}{5}}}{4998} \end {align*}
Given now that we have the solution \begin {align*} x \left (t \right )&=-\frac {\left (-1470 \,{\mathrm e}^{\frac {17 t}{5}}+833 \,{\mathrm e}^{\frac {12 t}{5}}-2142 \,{\mathrm e}^{\frac {7 t}{5}} t +102 \,{\mathrm e}^{\frac {7 t}{5}}-4998 c_{1} \right ) {\mathrm e}^{-\frac {7 t}{5}}}{4998} \tag {6} \end {align*}
Then substituting (6) into (3) gives \begin {align*} y \left (t \right )&=-\frac {{\mathrm e}^{2 t}}{2}+\frac {9 \,{\mathrm e}^{-\frac {7 t}{5}} {\mathrm e}^{\frac {7 t}{5}} t}{14}-\frac {{\mathrm e}^{-\frac {7 t}{5}} {\mathrm e}^{\frac {12 t}{5}}}{4}+\frac {15 \,{\mathrm e}^{-\frac {7 t}{5}} {\mathrm e}^{\frac {17 t}{5}}}{34}-\frac {3 \,{\mathrm e}^{-\frac {7 t}{5}} {\mathrm e}^{\frac {7 t}{5}}}{98}+\frac {3 c_{1} {\mathrm e}^{-\frac {7 t}{5}}}{2}+\frac {{\mathrm e}^{t}}{2}-\frac {t}{2}-\frac {1}{2} \tag {7} \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 51
dsolve([diff(x(t),t)+diff(y(t),t)+2*x(t)+y(t)=exp(2*t)+t,diff(x(t),t)+diff(y(t),t)-x(t)+3*y(t)=exp(t)-1],singsol=all)
\begin{align*} x \left (t \right ) &= \frac {3 t}{7}-\frac {1}{49}-\frac {{\mathrm e}^{t}}{6}+\frac {5 \,{\mathrm e}^{2 t}}{17}+{\mathrm e}^{-\frac {7 t}{5}} c_{1} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{17}+\frac {t}{7}-\frac {26}{49}+\frac {{\mathrm e}^{t}}{4}+\frac {3 \,{\mathrm e}^{-\frac {7 t}{5}} c_{1}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.185 (sec). Leaf size: 84
DSolve[{x'[t]+y'[t]+2*x[t]+y[t]==Exp[2*t]+t,x'[t]+y'[t]-x[t]+3*y[t]==Exp[t]-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {3 t}{7}-\frac {e^t}{6}+\frac {5 e^{2 t}}{17}+\frac {5}{72} c_1 e^{-7 t/5}-\frac {1}{49} \\ y(t)\to \frac {t}{7}+\frac {e^t}{4}-\frac {e^{2 t}}{17}+\frac {5}{48} c_1 e^{-7 t/5}-\frac {26}{49} \\ \end{align*}