Internal problem ID [8454]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 874.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (-256 a \,x^{2}+512+512 y^{2}+128 y a \,x^{4}+8 a^{2} x^{8}+512 y^{3}+192 y^{2} a \,x^{4}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(diff(y(x),x) = 1/512*(-256*a*x^2+512+512*y(x)^2+128*y(x)*a*x^4+8*a^2*x^8+512*y(x)^3+192*x^4*a*y(x)^2+24*y(x)*a^2*x^8+a^3*x^12)*x,y(x), singsol=all)
\[ y \relax (x ) = -\frac {a \,x^{4}}{8}-\frac {1}{3}+\frac {29 \RootOf \left (x^{2}-162 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+6 c_{1}\right )}{9} \]
✓ Solution by Mathematica
Time used: 0.175 (sec). Leaf size: 101
DSolve[y'[x] == (x*(512 - 256*a*x^2 + 8*a^2*x^8 + a^3*x^12 + 128*a*x^4*y[x] + 24*a^2*x^8*y[x] + 512*y[x]^2 + 192*a*x^4*y[x]^2 + 512*y[x]^3))/512,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {1}{8} \left (3 a x^5+8 x\right )+3 x y(x)}{\sqrt [3]{29} \sqrt [3]{x^3}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{18} 29^{2/3} \left (x^3\right )^{2/3}+c_1,y(x)\right ] \]