60.9.12 problem 1867

Internal problem ID [11866]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1867
Date solved : Monday, January 27, 2025 at 11:44:00 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+y \left (t \right )-t^{2}+6 t +1&=0\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )&=-3 t^{2}+3 t +1 \end{align*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 41

dsolve([diff(x(t),t)+y(t)-t^2+6*t+1=0,diff(y(t),t)-x(t)=-3*t^2+3*t+1],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} +3 t^{2}-t -13 \\ y \left (t \right ) &= t^{2}-c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )-12 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 200

DSolve[{D[x[t],t]+y[t]-t^2+6*t+1==0,D[y[t],t]-x[t]==-3*t^2+3*t+1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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