87.20.16 problem 17

Internal problem ID [23701]
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 : 17
Date solved : Thursday, October 02, 2025 at 09:44:24 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x-2 y \left (t \right )+2 t^{2}\\ y^{\prime }\left (t \right )&=5 x+y \left (t \right )-1 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&={\frac {534}{2197}} \\ y \left (0\right )&={\frac {567}{2197}} \\ \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 27

ode:=[diff(x(t),t) = 3*x(t)-2*y(t)+2*t^2, diff(y(t),t) = 5*x(t)+y(t)-1]; 
ic:=[x(0) = 534/2197, y(0) = 567/2197]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {534}{2197}-\frac {2}{13} t^{2}+\frac {36}{169} t \\ y \left (t \right ) &= \frac {10}{13} t^{2}+\frac {567}{2197}+\frac {80}{169} t \\ \end{align*}

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 36

ode={D[x[t],t]==3*x[t]-2*y[t]+2*t^2,D[y[t],t]==5*x[t]+y[t]-1}; 
ic={x[0]==534/2197,y[0]==567/2197}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {-338 t^2+468 t+534}{2197}\\ y(t)&\to \frac {1690 t^2+1040 t+567}{2197} \end{align*}

✓ Sympy. Time used: 1.663 (sec). Leaf size: 146

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*t**2 - 3*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - y(t) + Derivative(y(t), t) + 1,0)] 
ics = {x(0): 534/2197, y(0): 567/2197} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {2 t^{2} \sin ^{2}{\left (3 t \right )}}{13} - \frac {2 t^{2} \cos ^{2}{\left (3 t \right )}}{13} + \frac {36 t \sin ^{2}{\left (3 t \right )}}{169} + \frac {36 t \cos ^{2}{\left (3 t \right )}}{169} + \frac {534 \sin ^{2}{\left (3 t \right )}}{2197} + \frac {534 \cos ^{2}{\left (3 t \right )}}{2197}, \ y{\left (t \right )} = \frac {10 t^{2} \sin ^{2}{\left (3 t \right )}}{13} + \frac {10 t^{2} \cos ^{2}{\left (3 t \right )}}{13} + \frac {80 t \sin ^{2}{\left (3 t \right )}}{169} + \frac {80 t \cos ^{2}{\left (3 t \right )}}{169} + \frac {567 \sin ^{2}{\left (3 t \right )}}{2197} + \frac {567 \cos ^{2}{\left (3 t \right )}}{2197}\right ] \]