60.9.27 problem 1882

Internal problem ID [11881]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1882
Date solved : Tuesday, January 28, 2025 at 06:23:58 PM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-3 x \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.095 (sec). Leaf size: 117

dsolve([2*diff(x(t),t)+diff(y(t),t)-3*x(t)=0,diff(x(t),t$2)+diff(y(t),t)-2*y(t)=exp(2*t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{4}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{8}+c_{1} {\mathrm e}^{t}-\frac {7 c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{4}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{4}-\frac {7 c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{4}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.301 (sec). Leaf size: 932

DSolve[{2*D[x[t],t]+D[y[t],t]-3*x[t]==0,D[x[t],{t,2}]+D[y[t],t]-2*y[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{138} e^{t/2} \left (3 \left (23 e^{t/2}-3 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{138} e^{\frac {3 K[1]}{2}} \left (-23 \cos \left (\frac {1}{2} \sqrt {23} K[1]\right )+23 e^{-\frac {K[1]}{2}}-11 \sqrt {23} \sin \left (\frac {1}{2} \sqrt {23} K[1]\right )\right )dK[1]+\left (23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{138} e^{\frac {3 K[2]}{2}} \left (115 \cos \left (\frac {1}{2} \sqrt {23} K[2]\right )+23 e^{-\frac {K[2]}{2}}-17 \sqrt {23} \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )\right )dK[2]-2 \left (-23 e^{t/2}+\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{138} e^{\frac {3 K[3]}{2}} \left (-23 \cos \left (\frac {1}{2} \sqrt {23} K[3]\right )+23 e^{-\frac {K[3]}{2}}+25 \sqrt {23} \sin \left (\frac {1}{2} \sqrt {23} K[3]\right )\right )dK[3]+3 c_1 \left (23 e^{t/2}-3 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right )-2 c_3 \left (-23 e^{t/2}+\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right )+c_2 \left (23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right )\right ) \\ y(t)\to \frac {1}{138} e^{t/2} \left (\left (23 e^{t/2}-25 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{138} e^{\frac {3 K[2]}{2}} \left (115 \cos \left (\frac {1}{2} \sqrt {23} K[2]\right )+23 e^{-\frac {K[2]}{2}}-17 \sqrt {23} \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )\right )dK[2]+3 \left (23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{138} e^{\frac {3 K[1]}{2}} \left (-23 \cos \left (\frac {1}{2} \sqrt {23} K[1]\right )+23 e^{-\frac {K[1]}{2}}-11 \sqrt {23} \sin \left (\frac {1}{2} \sqrt {23} K[1]\right )\right )dK[1]+2 \left (23 e^{t/2}-4 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+46 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{138} e^{\frac {3 K[3]}{2}} \left (-23 \cos \left (\frac {1}{2} \sqrt {23} K[3]\right )+23 e^{-\frac {K[3]}{2}}+25 \sqrt {23} \sin \left (\frac {1}{2} \sqrt {23} K[3]\right )\right )dK[3]+c_2 \left (23 e^{t/2}-25 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right )+2 c_3 \left (23 e^{t/2}-4 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+46 \cos \left (\frac {\sqrt {23} t}{2}\right )\right )+3 c_1 \left (23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right )\right ) \\ \end{align*}