62.17.10 problem Ex 10

Internal problem ID [12910]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 28. Summary. Page 59
Problem number : Ex 10
Date solved : Tuesday, January 28, 2025 at 04:43:06 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.505 (sec). Leaf size: 87

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

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

✓ Solution by Mathematica

Time used: 1.130 (sec). Leaf size: 178

DSolve[y[x]==x*D[y[x],x]+y[x]*(D[y[x],x])^2/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 {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{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 {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{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*}