Internal problem ID [8842]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 507.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [`y=_G(x,y')`]
\[ \boxed {\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right )=0} \]
✓ Solution by Maple
Time used: 1.735 (sec). Leaf size: 207
dsolve((y(x)^4-a^2*x^2)*diff(y(x),x)^2+2*a^2*x*y(x)*diff(y(x),x)+y(x)^2*(y(x)^2-a^2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right )-\operatorname {RootOf}\left (c_{1} \sqrt {\operatorname {RootOf}\left (\left (-y \left (x \right )^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y \left (x \right )^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) \textit {\_Z}}+a \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {3}{4}\right ], \frac {\textit {\_Z}^{2} \left (2 \operatorname {RootOf}\left (\left (-y \left (x \right )^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y \left (x \right )^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) a^{2} x +\textit {\_Z}^{2}-a^{2}\right )}{\textit {\_Z}^{4}-a^{2} x^{2}}\right )+\textit {\_Z} \left (-\frac {a^{2} \left (2 \operatorname {RootOf}\left (\left (-y \left (x \right )^{4}+a^{2} x^{2}\right ) \textit {\_Z}^{2}-y \left (x \right )^{2}+a^{2}-2 \textit {\_Z} \,a^{2} x \right ) \textit {\_Z}^{2} x -\textit {\_Z}^{2}+x^{2}\right )}{\textit {\_Z}^{4}-a^{2} x^{2}}\right )^{\frac {1}{4}}\right ) &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 104.922 (sec). Leaf size: 395
DSolve[y[x]^2*(-a^2 + y[x]^2) + 2*a^2*x*y[x]*y'[x] + (-(a^2*x^2) + y[x]^4)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\left \{x&=\frac {a^2 K[1] y(K[1])-\sqrt {a^2 K[1]^2 \left (K[1]^2+1\right ) y(K[1])^4}}{a^2 K[1]^2},y(x)&=\frac {-4 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},-K[1]^2\right )-a c_1 \sqrt {K[1]}}{4 \sqrt [4]{K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \\ \text {Solve}\left [\left \{x&=\frac {a^2 K[1] y(K[1])-\sqrt {a^2 K[1]^2 \left (K[1]^2+1\right ) y(K[1])^4}}{a^2 K[1]^2},y(x)&=\frac {4 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},-K[1]^2\right )-a c_1 \sqrt {K[1]}}{4 \sqrt [4]{K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \\ \text {Solve}\left [\left \{x&=\frac {a^2 K[1] y(K[1])+\sqrt {a^2 K[1]^2 \left (K[1]^2+1\right ) y(K[1])^4}}{a^2 K[1]^2},y(x)&=\frac {-4 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},-K[1]^2\right )+a c_1 \sqrt {K[1]}}{4 \sqrt [4]{K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \\ \text {Solve}\left [\left \{x&=\frac {a^2 K[1] y(K[1])+\sqrt {a^2 K[1]^2 \left (K[1]^2+1\right ) y(K[1])^4}}{a^2 K[1]^2},y(x)&=\frac {4 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},-K[1]^2\right )+a c_1 \sqrt {K[1]}}{4 \sqrt [4]{K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \\ \end{align*}