90.35.10 problem 10

Internal problem ID [25511]
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 : 10
Date solved : Sunday, October 12, 2025 at 05:55:41 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=\frac {4 t y_{1} \left (t \right )}{t^{2}+1}+\frac {6 y_{2} \left (t \right ) t}{t^{2}+1}-3 t\\ \frac {d}{d t}y_{2} \left (t \right )&=-\frac {2 t y_{1} \left (t \right )}{t^{2}+1}-\frac {4 y_{2} \left (t \right ) t}{t^{2}+1}+t \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 161

ode:=[diff(y__1(t),t) = 4*t/(t^2+1)*y__1(t)+6*y__2(t)*t/(t^2+1)-3*t, diff(y__2(t),t) = -2*t/(t^2+1)*y__1(t)-4*y__2(t)*t/(t^2+1)+t]; 
ic:=[y__1(1) = 1, y__2(1) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= -\frac {6 \ln \left (t^{2}+1\right ) t^{4}-4 \left (\frac {3 \ln \left (2\right )}{2}-\frac {3}{4}\right ) t^{4}-3 t^{4}+12 \ln \left (t^{2}+1\right ) t^{2}-8 \left (\frac {3 \ln \left (2\right )}{2}-\frac {3}{4}\right ) t^{2}-6 t^{2}+6 \ln \left (t^{2}+1\right )-6 \ln \left (2\right )-8}{4 \left (t^{2}+1\right )} \\ y_{2} \left (t \right ) &= \frac {6 \ln \left (t^{2}+1\right ) t^{4}-4 \left (\frac {3 \ln \left (2\right )}{2}-\frac {3}{4}\right ) t^{4}-3 t^{4}+12 \ln \left (t^{2}+1\right ) t^{2}-8 \left (\frac {3 \ln \left (2\right )}{2}-\frac {3}{4}\right ) t^{2}-6 t^{2}+6 \ln \left (t^{2}+1\right )-6 \ln \left (2\right )-24}{12 t^{2}+12} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 92

ode={D[y1[t],t]==2*y1[t]*2*t/(1+t^2)+3*y2[t]*2*t/(1+t^2)-3*t, D[y2[t],t]==-y1[t]*2*t/(1+t^2)-2*y2[t]*2*t/(1+t^2)+t}; 
ic={y1[1]==1,y2[1]==-1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to \frac {t^4 \log (8)+t^2 \log (64)-3 \left (t^2+1\right )^2 \log \left (t^2+1\right )+4+\log (8)}{2 \left (t^2+1\right )}\\ \text {y2}(t)&\to -\frac {t^4 \log (2)+t^2 \log (4)-\left (t^2+1\right )^2 \log \left (t^2+1\right )+4+\log (2)}{2 \left (t^2+1\right )} \end{align*}

✓ Sympy. Time used: 0.395 (sec). Leaf size: 92

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(3*t - 4*t*y1(t)/(t**2 + 1) - 6*t*y2(t)/(t**2 + 1) + Derivative(y1(t), t),0),Eq(-t + 2*t*y1(t)/(t**2 + 1) + 4*t*y2(t)/(t**2 + 1) + Derivative(y2(t), t),0)] 
ics = {y1(1): 1, y2(1): -1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - \frac {3 t^{2} \log {\left (t^{2} + 1 \right )}}{2} + \frac {3 t^{2} \log {\left (2 \right )}}{2} - \frac {3 \log {\left (t^{2} + 1 \right )}}{2} + \frac {3 \log {\left (2 \right )}}{2} + \frac {2}{t^{2} + 1}, \ y_{2}{\left (t \right )} = \frac {t^{2} \log {\left (t^{2} + 1 \right )}}{2} - \frac {t^{2} \log {\left (2 \right )}}{2} + \frac {\log {\left (t^{2} + 1 \right )}}{2} - \frac {\log {\left (2 \right )}}{2} - \frac {2}{t^{2} + 1}\right ] \]