60.1.505 problem 508

Internal problem ID [10519]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 508
Date solved : Monday, January 27, 2025 at 08:44:32 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Solution by Maple

Time used: 0.217 (sec). Leaf size: 62

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= 0 \\ y &= -\operatorname {arctanh}\left (\operatorname {RootOf}\left (\operatorname {arctanh}\left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 \,\operatorname {arctanh}\left (\textit {\_Z} \right ) c_{1} \textit {\_Z}^{2}+c_{1}^{2} \textit {\_Z}^{2}+x^{2} \textit {\_Z}^{2}-x^{2}\right )\right )+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.334 (sec). Leaf size: 104

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

\[ \text {Solve}\left [-\frac {\sqrt {x^2+y(x)^2} y(x) \log (y(x))}{x^2 \sqrt {\frac {y(x)^2 \left (x^2+y(x)^2\right )}{x^4}}}+\frac {\sqrt {x^2+y(x)^2} y(x) \log \left (\sqrt {x^2+y(x)^2}+x\right )}{x^2 \sqrt {\frac {y(x)^2 \left (x^2+y(x)^2\right )}{x^4}}}+y(x)=c_1,y(x)\right ] \]