68.4.29 problem 29

Internal problem ID [17209]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 29
Date solved : Thursday, October 02, 2025 at 01:54:49 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 24

ode:=ln(x)^(1/2)/x = exp(3/y(x))/y(x)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {3}{\operatorname {RootOf}\left (2 \ln \left (x \right )^{{3}/{2}}-3 \,\operatorname {Ei}_{1}\left (\textit {\_Z} \right )+3 c_1 \right )} \]

✓ Mathematica. Time used: 0.306 (sec). Leaf size: 29

ode=Sqrt[Log[x]]/x==Exp[3/y[x]]/y[x]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\operatorname {ExpIntegralEi}\left (\frac {3}{\text {$\#$1}}\right )\&\right ]\left [\frac {2}{3} \log ^{\frac {3}{2}}(x)+c_1\right ] \end{align*}

✓ Sympy. Time used: 0.460 (sec). Leaf size: 19

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

\[ - \operatorname {Ei}{\left (\frac {3}{y{\left (x \right )}} \right )} = C_{1} + \frac {2 \log {\left (x \right )}^{\frac {3}{2}}}{3} \]