Internal problem ID [9148]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 813.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\[ \boxed {y^{\prime }-\frac {\left (-x^{3} \sqrt {a}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2}=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 38
dsolve(diff(y(x),x) = 1/2*(-a^(1/2)*x^3+2*(a*x^4+8*y(x))^(1/2)+2*x^2*(a*x^4+8*y(x))^(1/2)+2*x^3*(a*x^4+8*y(x))^(1/2))*a^(1/2),y(x), singsol=all)
\[ \frac {\sqrt {a \,x^{4}+8 y \left (x \right )}}{4}-\sqrt {a}\, \left (\frac {1}{4} x^{4}+\frac {1}{3} x^{3}+x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.872 (sec). Leaf size: 64
DSolve[y'[x] == (Sqrt[a]*(-(Sqrt[a]*x^3) + 2*Sqrt[a*x^4 + 8*y[x]] + 2*x^2*Sqrt[a*x^4 + 8*y[x]] + 2*x^3*Sqrt[a*x^4 + 8*y[x]]))/2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{72} a \left (9 x^8+24 x^7+16 x^6+72 x^5+(87-72 c_1) x^4-96 c_1 x^3+144 x^2-288 c_1 x+144 c_1{}^2\right ) \]