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 : Monday, January 27, 2025 at 09:31:55 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

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

dsolve(diff(y(x),x)+2*x*y(x)*(1+a*x*y(x)^2) = 0,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*}

✓ Solution by Mathematica

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

DSolve[D[y[x],x]+2 x y[x](1+ a x y[x]^2)==0,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*}