60.1.253 problem 258

Internal problem ID [10267]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 258
Date solved : Wednesday, March 05, 2025 at 08:50:38 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

\begin{align*} 2 x^{2} y y^{\prime }+y^{2}-2 x^{3}-x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 33

ode:=2*x^2*y(x)*diff(y(x),x)+y(x)^2-2*x^3-x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +x^{2}} \\ y &= -\sqrt {{\mathrm e}^{\frac {1}{x}} c_{1} +x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.336 (sec). Leaf size: 195

ode=2*x^2*y[x]*D[y[x],x]+y[x]^2-2*x^3-x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -e^{\left .\frac {1}{2}\right /x} \sqrt {2 \int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]+c_1} \\ y(x)\to e^{\left .\frac {1}{2}\right /x} \sqrt {2 \int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]+c_1} \\ y(x)\to -\sqrt {2} e^{\left .\frac {1}{2}\right /x} \sqrt {\int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]} \\ y(x)\to \sqrt {2} e^{\left .\frac {1}{2}\right /x} \sqrt {\int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]} \\ \end{align*}

✓ Sympy. Time used: 0.486 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**3 + 2*x**2*y(x)*Derivative(y(x), x) - x**2 + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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