60.9.12 problem 1867
Internal
problem
ID
[11791]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1867
Date
solved
:
Wednesday, March 05, 2025 at 03:07:04 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*}
✓ Maple. Time used: 0.065 (sec). Leaf size: 41
ode:=[diff(x(t),t)+y(t)-t^2+6*t+1 = 0, diff(y(t),t)-x(t) = -3*t^2+3*t+1];
dsolve(ode);
\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*}
✓ Mathematica. Time used: 0.032 (sec). Leaf size: 200
ode={D[x[t],t]+y[t]-t^2+6*t+1==0,D[y[t],t]-x[t]==-3*t^2+3*t+1};
ic={};
DSolve[{ode,ic},{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*}
✓ Sympy. Time used: 0.292 (sec). Leaf size: 105
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-t**2 + 6*t + y(t) + Derivative(x(t), t) + 1,0),Eq(3*t**2 - 3*t - x(t) + Derivative(y(t), t) - 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} \sin {\left (t \right )} - C_{2} \cos {\left (t \right )} + 3 t^{2} \sin ^{2}{\left (t \right )} + 3 t^{2} \cos ^{2}{\left (t \right )} - t \sin ^{2}{\left (t \right )} - t \cos ^{2}{\left (t \right )} - 13 \sin ^{2}{\left (t \right )} - 13 \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + t^{2} \sin ^{2}{\left (t \right )} + t^{2} \cos ^{2}{\left (t \right )} - 12 t \sin ^{2}{\left (t \right )} - 12 t \cos ^{2}{\left (t \right )}\right ]
\]