29.4.2 problem 87

Internal problem ID [4693]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 87
Date solved : Tuesday, March 04, 2025 at 07:04:26 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+2 x y \left (1+a x y^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 75

ode:=diff(y(x),x)+2*x*y(x)*(1+a*x*y(x)^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= -\frac {2}{\sqrt {\left (\sqrt {\pi }\, a \,\operatorname {erf}\left (\sqrt {2}\, x \right ) \sqrt {2}+4 c_{1} \right ) {\mathrm e}^{2 x^{2}}-4 a x}} \\ y \left (x \right ) &= \frac {2}{\sqrt {\left (\sqrt {\pi }\, a \,\operatorname {erf}\left (\sqrt {2}\, x \right ) \sqrt {2}+4 c_{1} \right ) {\mathrm e}^{2 x^{2}}-4 a x}} \\ \end{align*}

✓ Mathematica. Time used: 7.779 (sec). Leaf size: 106

ode=D[y[x],x]+2 x y[x](1+ a x y[x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {2}{\sqrt {\sqrt {2 \pi } a e^{2 x^2} \text {erf}\left (\sqrt {2} x\right )-4 a x+4 c_1 e^{2 x^2}}} \\ y(x)\to \frac {2}{\sqrt {\sqrt {2 \pi } a e^{2 x^2} \text {erf}\left (\sqrt {2} x\right )-4 a x+4 c_1 e^{2 x^2}}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 2.364 (sec). Leaf size: 100

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(2*x*(a*x*y(x)**2 + 1)*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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