Internal problem ID [2271]
Internal file name [OUTPUT/2271_Tuesday_February_27_2024_08_23_53_AM_59055855/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 26, page 115
Problem number: 6.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} 0&=-5 y^{\prime }\left (t \right )+3 x^{\prime }\left (t \right )+5 y \left (t \right )+5 t\\ 0&=-3 x^{\prime }\left (t \right )+5 y^{\prime }\left (t \right )+2 x \left (t \right ) \end {align*}
The system is \begin {align*} 5 y^{\prime }\left (t \right )-3 x^{\prime }\left (t \right )&=5 y \left (t \right )+5 t\tag {1}\\ 3 x^{\prime }\left (t \right )-5 y^{\prime }\left (t \right )&=2 x \left (t \right )\tag {2} \end {align*}
Since the left side is the same, this implies \begin {align*} 5 y \left (t \right )+5 t&=2 x \left (t \right )\\ y \left (t \right )&=\frac {2 x \left (t \right )}{5}-t\tag {3} \end {align*}
Taking derivative of the above w.r.t. \(t\) gives \begin {align*} y^{\prime }\left (t \right )&=\frac {2 x^{\prime }\left (t \right )}{5}-1\tag {4} \end {align*}
Substituting (3,4) in (1) to eliminate \(y \left (t \right ),y^{\prime }\left (t \right )\) gives \begin {align*} -x^{\prime }\left (t \right )-5 &= 2 x \left (t \right )\\ x^{\prime }\left (t \right ) &= -2 x \left (t \right )-5\tag {5} \end {align*}
Which is now solved for \(x \left (t \right )\). Integrating both sides gives \begin {align*} \int \frac {1}{-2 x -5}d x &= t +c_{1}\\ -\frac {\ln \left (x +\frac {5}{2}\right )}{2}&=t +c_{1} \end {align*}
Solving for \(x\) gives these solutions \begin {align*} x_1&={\mathrm e}^{-2 t -2 c_{1}}-\frac {5}{2}\\ &=\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}-\frac {5}{2} \end {align*}
Given now that we have the solution \begin {align*} x \left (t \right )&=\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}-\frac {5}{2} \tag {6} \end {align*}
Then substituting (6) into (3) gives \begin {align*} y \left (t \right )&=-\frac {\left (5 t \,{\mathrm e}^{2 t} c_{1}^{2}+5 \,{\mathrm e}^{2 t} c_{1}^{2}-2\right ) {\mathrm e}^{-2 t}}{5 c_{1}^{2}} \tag {7} \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 28
dsolve([5*diff(y(t),t)-3*diff(x(t),t)-5*y(t)=5*t,3*diff(x(t),t)-5*diff(y(t),t)-2*x(t)=0],singsol=all)
\begin{align*} x \left (t \right ) &= \frac {5}{2}+{\mathrm e}^{\frac {2 t}{5}} c_{1} \\ y \left (t \right ) &= -1-\frac {2 \,{\mathrm e}^{\frac {2 t}{5}} c_{1}}{5}-t \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.137 (sec). Leaf size: 43
DSolve[{5*y'[t]-3*x'[t]-5*y[t]==5*t,3*x'[t]-5*y'[t]-2*x[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {5}{6} \left (3+2 c_1 e^{2 t/5}\right ) \\ y(t)\to -t-\frac {2}{3} c_1 e^{2 t/5}-1 \\ \end{align*}