83.23.8 problem 8

Internal problem ID [19289]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 01:24:29 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 3 y&=2 x y^{\prime }-\frac {2 {y^{\prime }}^{2}}{x} \end{align*}

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 77

dsolve(3*y(x)=2*diff(y(x),x)*x-2*diff(y(x),x)^2/x,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.627 (sec). Leaf size: 146

DSolve[3*y[x]==2*D[y[x],x]*x-2*D[y[x],x]^2/x,y[x],x,IncludeSingularSolutions -> True]
 

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