60.1.484 problem 487

Internal problem ID [10498]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 487
Date solved : Monday, January 27, 2025 at 08:21:21 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.150 (sec). Leaf size: 105

dsolve(y(x)^2*diff(y(x),x)^2-6*x^3*diff(y(x),x)+4*x^2*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= \frac {18^{{1}/{3}} x^{{4}/{3}}}{2} \\ y &= -\frac {18^{{1}/{3}} x^{{4}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {18^{{1}/{3}} x^{{4}/{3}} \left (i \sqrt {3}-1\right )}{4} \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} \right )+4 c_{1} \right ) x^{{4}/{3}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.720 (sec). Leaf size: 235

DSolve[4*x^2*y[x] - 6*x^3*D[y[x],x] + y[x]^2*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {3}{4} \log (y(x))-\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \text {arctanh}\left (\frac {3 x^2}{\sqrt {9 x^4-4 y(x)^3}+2 i y(x)^{3/2}}\right )}{x \sqrt {9 x^4-4 y(x)^3}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {3}{4} \log (y(x))+\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \text {arctanh}\left (\frac {3 x^2}{\sqrt {9 x^4-4 y(x)^3}+2 i y(x)^{3/2}}\right )}{x \sqrt {9 x^4-4 y(x)^3}}&=c_1,y(x)\right ] \\ y(x)\to \left (-\frac {3}{2}\right )^{2/3} x^{4/3} \\ y(x)\to \left (\frac {3}{2}\right )^{2/3} x^{4/3} \\ y(x)\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} x^{4/3} \\ \end{align*}