60.9.11 problem 1866

Internal problem ID [11790]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1866
Date solved : Sunday, March 30, 2025 at 09:15:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+2 y \left (t \right )&=3 t\\ \frac {d}{d t}y \left (t \right )-2 x \left (t \right )&=4 \end{align*}

✓ Maple. Time used: 0.145 (sec). Leaf size: 38

ode:={diff(x(t),t)+2*y(t) = 3*t, diff(y(t),t)-2*x(t) = 4}; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 -\frac {5}{4} \\ y \left (t \right ) &= -\cos \left (2 t \right ) c_2 +\sin \left (2 t \right ) c_1 +\frac {3 t}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 156

ode={D[x[t],t]+2*y[t]==3*t,D[y[t],t]-2*x[t]==4}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \cos (2 t) \int _1^t(3 \cos (2 K[1]) K[1]+4 \sin (2 K[1]))dK[1]-\sin (2 t) \int _1^t(4 \cos (2 K[2])-3 K[2] \sin (2 K[2]))dK[2]+c_1 \cos (2 t)-c_2 \sin (2 t) \\ y(t)\to \cos (2 t) \int _1^t(4 \cos (2 K[2])-3 K[2] \sin (2 K[2]))dK[2]+\sin (2 t) \int _1^t(3 \cos (2 K[1]) K[1]+4 \sin (2 K[1]))dK[1]+c_2 \cos (2 t)+c_1 \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*t + 2*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + Derivative(y(t), t) - 4,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} \sin {\left (2 t \right )} - C_{2} \cos {\left (2 t \right )} - \frac {5 \sin ^{2}{\left (2 t \right )}}{4} - \frac {5 \cos ^{2}{\left (2 t \right )}}{4}, \ y{\left (t \right )} = C_{1} \cos {\left (2 t \right )} - C_{2} \sin {\left (2 t \right )} + \frac {3 t \sin ^{2}{\left (2 t \right )}}{2} + \frac {3 t \cos ^{2}{\left (2 t \right )}}{2}\right ] \]