20.4.31 problem Problem 47

Internal problem ID [3666]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 47
Date solved : Monday, January 27, 2025 at 07:53:00 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }-\frac {y}{2 x \ln \left (x \right )}&=2 x y^{3} \end{align*}

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 86

dsolve(diff(y(x),x)-1/(2*x*ln(x))*y(x)=2*x*y(x)^3,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {\sqrt {-2 \ln \left (x \right )^{2} x^{2}+\left (x^{2}+c_{1} \right ) \ln \left (x \right )}}{2 \ln \left (x \right ) x^{2}-x^{2}-c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {-2 \ln \left (x \right )^{2} x^{2}+\left (x^{2}+c_{1} \right ) \ln \left (x \right )}}{2 \ln \left (x \right ) x^{2}-x^{2}-c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.255 (sec). Leaf size: 63

DSolve[D[y[x],x]-1/(2*x*Log[x])*y[x]==2*x*y[x]^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {\log (x)}}{\sqrt {x^2-2 x^2 \log (x)+c_1}} \\ y(x)\to \frac {\sqrt {\log (x)}}{\sqrt {x^2-2 x^2 \log (x)+c_1}} \\ y(x)\to 0 \\ \end{align*}