69.31.7 problem 821

Internal problem ID [18561]
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 : Thursday, October 02, 2025 at 03:14:55 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.138 (sec). Leaf size: 31

ode:=[diff(x(t),t)+y(t) = t^2, diff(y(t),t)-x(t) = t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +t \\ y \left (t \right ) &= t^{2}-\cos \left (t \right ) c_2 +\sin \left (t \right ) c_1 -1 \\ \end{align*}

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 126

ode={D[x[t],t]+y[t]==t^2,D[y[t],t]-x[t]==t}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t**2 + y(t) + Derivative(x(t), t),0),Eq(-t - x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} \sin {\left (t \right )} - C_{2} \cos {\left (t \right )} + t \sin ^{2}{\left (t \right )} + t \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + t^{2} \sin ^{2}{\left (t \right )} + t^{2} \cos ^{2}{\left (t \right )} - \sin ^{2}{\left (t \right )} - \cos ^{2}{\left (t \right )}\right ] \]