77.1.78 problem 97 (page 135)

Internal problem ID [17897]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 97 (page 135)
Date solved : Monday, March 31, 2025 at 04:49:10 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 81

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

\begin{align*} y &= -\frac {x^{2}}{2} \\ y &= -\frac {1}{4} x^{2}+\frac {1}{2} c_1 x +\frac {1}{4} c_1^{2} \\ y &= -\frac {1}{4} x^{2}-\frac {1}{2} c_1 x +\frac {1}{4} c_1^{2} \\ y &= -\frac {1}{4} x^{2}-\frac {1}{2} c_1 x +\frac {1}{4} c_1^{2} \\ y &= -\frac {1}{4} x^{2}+\frac {1}{2} c_1 x +\frac {1}{4} c_1^{2} \\ \end{align*}

✗ Mathematica

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

Timed out

✓ Sympy. Time used: 1.633 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = - \frac {x^{2}}{2} + \frac {\left (C_{1} + x\right )^{2}}{4} \]