60.9.29 problem 1884

Internal problem ID [11883]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1884
Date solved : Tuesday, January 28, 2025 at 06:23:59 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )+2 y \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}x \left (t \right )-2 \frac {d}{d t}y \left (t \right )&=2 t -\cos \left (2 t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.273 (sec). Leaf size: 68

dsolve([diff(x(t),t)-x(t)+2*y(t)=0,diff(x(t),t,t)-2*diff(y(t),t)=2*t-cos(2*t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= -t^{2}+2 c_{1} {\mathrm e}^{\frac {t}{2}}+\frac {2 \cos \left (2 t \right )}{17}+\frac {\sin \left (2 t \right )}{34}-4 t +c_{2} \\ y \left (t \right ) &= -t +\frac {c_{1} {\mathrm e}^{\frac {t}{2}}}{2}+\frac {9 \sin \left (2 t \right )}{68}+\frac {\cos \left (2 t \right )}{34}+2-\frac {t^{2}}{2}+\frac {c_{2}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.426 (sec). Leaf size: 199

DSolve[{D[x[t],t]-x[t]+2*y[t]==0,D[x[t],{t,2}]-2*D[y[t],t]==2*t-Cos[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 8 \int _1^t-\frac {1}{8} e^{-\frac {K[1]}{2}} \left (e^{t/2}-8 e^{\frac {K[1]}{2}}\right ) (\cos (2 K[1])-2 K[1])dK[1]+7 \int _1^t(2 K[2]-\cos (2 K[2]))dK[2]+8 c_1 e^{t/2}+8 c_2 e^{t/2}-c_2 \\ y(t)\to 2 \left (\int _1^t-\frac {1}{8} e^{-\frac {K[1]}{2}} \left (e^{t/2}-8 e^{\frac {K[1]}{2}}\right ) (\cos (2 K[1])-2 K[1])dK[1]+(c_1+c_2) e^{t/2}-c_2\right )+\frac {3}{2} \left (\int _1^t(2 K[2]-\cos (2 K[2]))dK[2]+c_2\right ) \\ \end{align*}