83.22.13 problem 13

Internal problem ID [19267]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 01:17:55 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (y-x y^{\prime }\right )&=y {y^{\prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 87

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

\begin{align*} y \left (x \right ) &= -\frac {i x^{2}}{2} \\ y \left (x \right ) &= \frac {i x^{2}}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{4} \\ y \left (x \right ) &= \frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{4} \\ y \left (x \right ) &= -\frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.687 (sec). Leaf size: 169

DSolve[x^2*(y[x]-D[y[x],x]*x)==y[x]*D[y[x],x]^2,y[x],x,IncludeSingularSolutions -> True]
 

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