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 : Tuesday, March 04, 2025 at 05:04:36 PM
CAS classification : [_Bernoulli]

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

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

ode:=diff(y(x),x)-1/2*y(x)/x/ln(x) = 2*x*y(x)^3; 
dsolve(ode,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*}

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

ode=D[y[x],x]-1/(2*x*Log[x])*y[x]==2*x*y[x]^3; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.550 (sec). Leaf size: 46

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

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