62.39.1 problem Ex 1

Internal problem ID [13024]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter X, System of simulataneous equations. Article 64. Systems of linear equations with constant coefficients. Page 150
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:49:30 AM
CAS classification : system_of_ODEs

\begin{align*} 3 \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+2 y \left (t \right )&={\mathrm e}^{t}\\ 4 x \left (t \right )-3 \frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=3 t \end{align*}

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 46

dsolve([3*diff(x(t),t)+3*x(t)+2*y(t)=exp(t),4*x(t)-3*diff(y(t),t)+3*y(t)=3*t],singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{3}} c_{2}}{2}-{\mathrm e}^{-\frac {t}{3}} c_{1} -6 t \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{3}} c_{2} +{\mathrm e}^{-\frac {t}{3}} c_{1} +9 t +9+\frac {{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.071 (sec). Leaf size: 337

DSolve[{3*D[x[t],t]+3*x[t]+2*y[t]==Exp[t],4*x[t]-3*D[y[t],t]+3*y[t]==3*t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t/3} \left (-\left (e^{2 t/3}-2\right ) \int _1^t\frac {1}{3} e^{-\frac {K[1]}{3}} \left (-3 e^{\frac {2 K[1]}{3}} K[1]+3 K[1]-e^{K[1]}+2 e^{\frac {5 K[1]}{3}}\right )dK[1]-\left (e^{2 t/3}-1\right ) \int _1^t\frac {1}{3} e^{-\frac {K[2]}{3}} \left (3 e^{\frac {2 K[2]}{3}} K[2]-6 K[2]+2 e^{K[2]}-2 e^{\frac {5 K[2]}{3}}\right )dK[2]+c_1 \left (-e^{2 t/3}\right )-c_2 e^{2 t/3}+2 c_1+c_2\right ) \\ y(t)\to e^{-t/3} \left (2 \left (e^{2 t/3}-1\right ) \int _1^t\frac {1}{3} e^{-\frac {K[1]}{3}} \left (-3 e^{\frac {2 K[1]}{3}} K[1]+3 K[1]-e^{K[1]}+2 e^{\frac {5 K[1]}{3}}\right )dK[1]+\left (2 e^{2 t/3}-1\right ) \int _1^t\frac {1}{3} e^{-\frac {K[2]}{3}} \left (3 e^{\frac {2 K[2]}{3}} K[2]-6 K[2]+2 e^{K[2]}-2 e^{\frac {5 K[2]}{3}}\right )dK[2]+2 c_1 e^{2 t/3}+2 c_2 e^{2 t/3}-2 c_1-c_2\right ) \\ \end{align*}