22.1.69 problem 72

Internal problem ID [4375]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 72
Date solved : Tuesday, September 30, 2025 at 07:23:42 AM
CAS classification : [_Bernoulli]

\begin{align*} 3 x y^{\prime }-3 x y^{4} \ln \left (x \right )-y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 160

ode:=3*x*diff(y(x),x)-3*x*y(x)^4*ln(x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{2}/{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1 \right )^{2}\right )}^{{1}/{3}}}{6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1} \\ y &= -\frac {2^{{2}/{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1 \right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 c_1} \\ y &= \frac {2^{{2}/{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_1 \right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 120

ode=3*x*D[y[x],x]-3*x*y[x]^4*Log[x]-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {(-2)^{2/3} \sqrt [3]{x}}{\sqrt [3]{3 x^2-6 x^2 \log (x)+4 c_1}}\\ y(x)&\to \frac {2^{2/3} \sqrt [3]{x}}{\sqrt [3]{3 x^2-6 x^2 \log (x)+4 c_1}}\\ y(x)&\to -\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x}}{\sqrt [3]{3 x^2-6 x^2 \log (x)+4 c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.584 (sec). Leaf size: 109

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*y(x)**4*log(x) + 3*x*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = 2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {x}{C_{1} + 6 x^{2} \log {\left (x \right )} - 3 x^{2}}} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]