89.33.36 problem 39

Internal problem ID [25020]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 39
Date solved : Thursday, October 02, 2025 at 11:47:19 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} {y^{\prime \prime }}^{2}-2 y^{\prime \prime }+{y^{\prime }}^{2}-2 x y^{\prime }+x^{2}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.286 (sec). Leaf size: 23

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

\begin{align*} y &= \frac {\left (x +1\right )^{2}}{2} \\ y &= \frac {x^{2}}{2}+\sin \left (x \right )+\frac {1}{2} \\ \end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2 - 2*x*Derivative(y(x), x) + Derivative(y(x), x)**2 + Derivative(y(x), (x, 2))**2 - 2*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1/2, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out