77.1.158 problem 185 (page 297)

Internal problem ID [18048]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 185 (page 297)
Date solved : Tuesday, January 28, 2025 at 11:23:27 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.249 (sec). Leaf size: 67

dsolve([diff(x(t),t)+x(t)+y(t)=t^2,diff(y(t),t)+y(t)+z(t)=2*t,diff(z(t),t)+z(t)=t],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 76

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

\begin{align*} x(t)\to t^2-3 t+\frac {1}{2} e^{-t} t (c_3 t-2 c_2)+c_1 e^{-t}+3 \\ y(t)\to e^{-t} \left (e^t t-c_3 t+c_2\right ) \\ z(t)\to t+c_3 e^{-t}-1 \\ \end{align*}