75.7.18 problem 193

Internal problem ID [16811]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 193
Date solved : Tuesday, January 28, 2025 at 09:33:28 AM
CAS classification : [_Bernoulli]

\begin{align*} x^{4} \ln \left (x \right )-2 x y^{3}+3 x^{2} y^{2} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 82

dsolve(( x^4*ln(x)-2*x*y(x)^3)+(3*x^2*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.703 (sec). Leaf size: 77

DSolve[( x^4*Log[x]-2*x*y[x]^3)+(3*x^2*y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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