85.83.10 problem 4 (a)

Internal problem ID [23007]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 444
Problem number : 4 (a)
Date solved : Thursday, October 02, 2025 at 09:17:28 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.073 (sec). Leaf size: 29

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

\begin{align*} x \left (t \right ) &= \frac {4 t^{{3}/{2}}}{9}+\frac {{\mathrm e}^{t}}{3}+c_2 \\ y \left (t \right ) &= \frac {2 t^{{3}/{2}}}{9}-\frac {{\mathrm e}^{t}}{3}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 46

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

\begin{align*} x(t)&\to \frac {4 t^{3/2}}{9}+\frac {e^t}{3}+c_1\\ y(t)&\to \frac {2 t^{3/2}}{9}-\frac {e^t}{3}+c_2 \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 34

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

\[ \left [ x{\left (t \right )} = C_{1} + \frac {4 t^{\frac {3}{2}}}{9} + \frac {e^{t}}{3}, \ y{\left (t \right )} = C_{2} + \frac {2 t^{\frac {3}{2}}}{9} - \frac {e^{t}}{3}\right ] \]