87.20.15 problem 16

Internal problem ID [23700]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 161
Problem number : 16
Date solved : Sunday, October 12, 2025 at 05:55:15 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (1\right )&=1 \\ y \left (1\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 35

ode:=[t*diff(x(t),t) = 3*x(t)-2*y(t), t*diff(y(t),t) = x(t)+y(t)-t^2]; 
ic:=[x(1) = 1, y(1) = 1/2]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 33

ode={t*D[x[t],t]==3*x[t]-2*y[t],t*D[y[t],t]==x[t]+y[t]-t^2}; 
ic={x[1]==1,y[1]==1/2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -t^2 (\cos (\log (t))-2)\\ y(t)&\to -\frac {1}{2} t^2 (\sin (\log (t))+\cos (\log (t))-2) \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 76

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

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