75.27.11 problem 786

Internal problem ID [17168]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 786
Date solved : Friday, March 14, 2025 at 04:49:26 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 18

ode:=[diff(diff(x(t),t),t) = x(t)^2+y(t), diff(y(t),t) = -2*x(t)*diff(x(t),t)+x(t)]; 
ic:=x(0) = 1D(x)(0) = 1y(0) = 0; 
dsolve([ode,ic]);
 

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

✗ Mathematica

ode={D[x[t],{t,2}]==x[t]^2+y[t],D[y[t],t]==-2*x[t]*D[x[t],t]+x[t]}; 
ic={x[0]==1,Derivative[1][x][0 ]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

ValueError : It solves only those systems of equations whose orders are equal