86.12.1 problem 1

Internal problem ID [23208]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 10. The Laplace transform. Exercise 10c at page 156
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:24:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+y \left (t \right )&=4\\ x \left (t \right )-\frac {d}{d t}y \left (t \right )&=3 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.070 (sec). Leaf size: 27

ode:=[diff(x(t),t)+y(t) = 4, x(t)-diff(y(t),t) = 3]; 
ic:=[x(0) = 0, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 4 \sin \left (t \right )-3 \cos \left (t \right )+3 \\ y \left (t \right ) &= -4 \cos \left (t \right )-3 \sin \left (t \right )+4 \\ \end{align*}

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 28

ode={D[x[t],t]+y[t]==4,x[t]-D[y[t],t]==3}; 
ic={x[0]==0,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 4 \sin (t)-3 \cos (t)+3\\ y(t)&\to -3 \sin (t)-4 \cos (t)+4 \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(y(t) + Derivative(x(t), t) - 4,0),Eq(x(t) - Derivative(y(t), t) - 3,0)] 
ics = {x(0): 0, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 3 \sin ^{2}{\left (t \right )} + 4 \sin {\left (t \right )} + 3 \cos ^{2}{\left (t \right )} - 3 \cos {\left (t \right )}, \ y{\left (t \right )} = 4 \sin ^{2}{\left (t \right )} - 3 \sin {\left (t \right )} + 4 \cos ^{2}{\left (t \right )} - 4 \cos {\left (t \right )}\right ] \]