28.1.69 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 : Monday, January 27, 2025 at 09:09:30 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 160

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

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

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 120

DSolve[3*x*D[y[x],x]-3*x*y[x]^4*Log[x]-y[x]==0,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*}