71.1.70 problem 90 (a)(page 123)

Internal problem ID [19246]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 90 (a)(page 123)
Date solved : Thursday, October 02, 2025 at 03:51:30 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 20

ode:=diff(y(x),x) = -x+(x^2+2*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ x -\sqrt {x^{2}+2 y}-c_1 = 0 \]

✓ Mathematica. Time used: 0.752 (sec). Leaf size: 38

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

\begin{align*} y(x)&\to \frac {1}{2} \left (1+e^{c_1}\right ) \left (2 x+1+e^{c_1}\right )\\ y(x)&\to 0\\ y(x)&\to x+\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.449 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \frac {C_{1} \left (C_{1} + 2 x\right )}{2} \]