29.13.13 problem 367

Internal problem ID [4967]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 13
Problem number : 367
Date solved : Monday, January 27, 2025 at 10:00:10 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 6 x^{3} y^{\prime }&=4 x^{2} y+\left (1-3 x \right ) y^{4} \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 127

dsolve(6*x^3*diff(y(x),x) = 4*x^2*y(x)+(1-3*x)*y(x)^4,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.215 (sec). Leaf size: 99

DSolve[6 x^3 D[y[x],x]==4 x^2 y[x]+(1-3 x)y[x]^4,y[x],x,IncludeSingularSolutions -> True]
 

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