75.5.6 problem 105

Internal problem ID [16671]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 5. Homogeneous equations. Exercises page 44
Problem number : 105
Date solved : Monday, March 31, 2025 at 03:04:42 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

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

\[ y = \frac {x \left (\ln \left (x \right )+c_1 -2\right )}{\ln \left (x \right )+c_1} \]

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 29

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

\begin{align*} y(x)\to \frac {x (\log (x)-2+2 c_1)}{\log (x)+2 c_1} \\ y(x)\to x \\ \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = x \left (1 - 16 x^{2}\right ) \]