75.31.7 problem 821

Internal problem ID [17274]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23.2 The method of undetermined coefficients. Exercises page 239
Problem number : 821
Date solved : Tuesday, January 28, 2025 at 09:58:44 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 126

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

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