77.1.152 problem 179 (page 297)

Internal problem ID [17971]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 179 (page 297)
Date solved : Monday, March 31, 2025 at 04:52:51 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y \left (0\right ) = 1\\ z \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 11

ode:=[diff(y(x),x) = -z(x), diff(z(x),x) = y(x)]; 
ic:=y(0) = 1z(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} y \left (x \right ) &= \cos \left (x \right ) \\ z \left (x \right ) &= \sin \left (x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 12

ode={D[y[x],x]==-z[x],D[z[x],x]==y[x]}; 
ic={y[0]==1,z[0]==0}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \cos (x) \\ z(x)\to \sin (x) \\ \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = - C_{1} \sin {\left (x \right )} - C_{2} \cos {\left (x \right )}, \ z{\left (x \right )} = C_{1} \cos {\left (x \right )} - C_{2} \sin {\left (x \right )}\right ] \]