67.6.2 problem Problem 4(b)

Internal problem ID [14035]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6.4 Reduction to a single ODE. Problems page 415
Problem number : Problem 4(b)
Date solved : Monday, March 31, 2025 at 08:22:34 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.137 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{\frac {t}{3}} c_1 -\frac {t^{2}}{2}-4 t +c_2 \\ y \left (t \right ) &= -\frac {3 \,{\mathrm e}^{\frac {t}{3}} c_1}{2}+2 t -6+\frac {t^{2}}{2}-c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.293 (sec). Leaf size: 208

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

\begin{align*} x(t)\to \left (2 e^{t/3}-1\right ) \int _1^t\frac {1}{3} \left (-3+4 e^{-\frac {K[1]}{3}}\right ) K[1]dK[1]+2 \left (e^{t/3}-1\right ) \int _1^t\left (K[2]-\frac {2}{3} e^{-\frac {K[2]}{3}} K[2]\right )dK[2]+c_1 \left (2 e^{t/3}-1\right )+2 c_2 \left (e^{t/3}-1\right ) \\ y(t)\to -\left (e^{t/3}-1\right ) \int _1^t\frac {1}{3} \left (-3+4 e^{-\frac {K[1]}{3}}\right ) K[1]dK[1]-\left (e^{t/3}-2\right ) \int _1^t\left (K[2]-\frac {2}{3} e^{-\frac {K[2]}{3}} K[2]\right )dK[2]+c_1 \left (-e^{t/3}\right )-c_2 e^{t/3}+c_1+2 c_2 \\ \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 44

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

\[ \left [ x{\left (t \right )} = - C_{1} - 2 C_{2} e^{\frac {t}{3}} - \frac {t^{2}}{2} - 4 t - 12, \ y{\left (t \right )} = C_{1} + C_{2} e^{\frac {t}{3}} + \frac {t^{2}}{2} + 2 t + 6\right ] \]