90.35.6 problem 6

Internal problem ID [25507]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 9. Linear Systems of Differential Equations. Exercises at page 719
Problem number : 6
Date solved : Sunday, October 12, 2025 at 05:55:40 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=\left (1+2 t \right ) y_{1} \left (t \right )+2 t y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-2 t y_{1} \left (t \right )+\left (1-2 t \right ) y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=1 \\ y_{2} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 22

ode:=[diff(y__1(t),t) = (2*t+1)*y__1(t)+2*t*y__2(t), diff(y__2(t),t) = -2*t*y__1(t)+(1-2*t)*y__2(t)]; 
ic:=[y__1(0) = 1, y__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{t} \left (t^{2}+1\right ) \\ y_{2} \left (t \right ) &= -{\mathrm e}^{t} t^{2} \\ \end{align*}

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 25

ode={D[y1[t],t]==(1+2*t)*y1[t]+2*t*y2[t], D[y2[t],t]==-2*t*y1[t]+(1-2*t)*y2[t]}; 
ic={y1[0]==1,y2[0]==0}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^t \left (t^2+1\right )\\ \text {y2}(t)&\to -e^t t^2 \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-2*t*y2(t) - (2*t + 1)*y1(t) + Derivative(y1(t), t),0),Eq(2*t*y1(t) - (1 - 2*t)*y2(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 1, y2(0): 0} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = t^{2} e^{t} + e^{t}, \ y_{2}{\left (t \right )} = - t^{2} e^{t}\right ] \]