54.1.28 problem 28

Internal problem ID [11342]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 28
Date solved : Tuesday, September 30, 2025 at 07:57:23 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 51

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

\[ y = \frac {\operatorname {erf}\left (\frac {x^{2}}{2}\right ) \sqrt {\pi }\, c_1 \,x^{2}+x^{2} \sqrt {\pi }+2 \,{\mathrm e}^{-\frac {x^{4}}{4}} c_1}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\frac {x^{2}}{2}\right ) c_1 +1\right )} \]

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to \frac {e \sqrt {\pi } x^2 \text {erf}\left (\frac {x^2}{2}\right )+2 e^{1-\frac {x^4}{4}}+2 c_1 x^2}{e \sqrt {\pi } \text {erf}\left (\frac {x^2}{2}\right )+2 c_1}\\ y(x)&\to x^2 \end{align*}

✓ Sympy. Time used: 5.019 (sec). Leaf size: 92

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

\[ y{\left (x \right )} = \frac {4 x^{6} e^{\frac {x^{4}}{4}} + 32 x^{2} e^{\frac {x^{4}}{4}} - 3 x^{2} e^{\frac {x^{4}}{2}} \operatorname {Ei}{\left (- \frac {x^{4}}{4} \right )} - 16 e^{\frac {x^{4}}{4}}}{4 x^{4} e^{\frac {x^{4}}{4}} + 32 e^{\frac {x^{4}}{4}} - 3 e^{\frac {x^{4}}{2}} \operatorname {Ei}{\left (- \frac {x^{4}}{4} \right )}} \]