60.2.84 problem 660

Internal problem ID [10671]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 660
Date solved : Monday, January 27, 2025 at 09:23:33 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x}{2}-\frac {a}{2}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y} \end{align*}

✓ Solution by Maple

Time used: 0.168 (sec). Leaf size: 29

dsolve(diff(y(x),x) = -1/2*x-1/2*a+x^2*(x^2+2*a*x+a^2+4*y(x))^(1/2),y(x), singsol=all)
 

\[ c_{1} +\frac {2 x^{3}}{3}-\sqrt {a^{2}+2 a x +x^{2}+4 y} = 0 \]

✓ Solution by Mathematica

Time used: 0.542 (sec). Leaf size: 42

DSolve[D[y[x],x] == -1/2*a - x/2 + x^2*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{36} \left (-9 a^2-18 a x+4 x^6-24 c_1 x^3-9 x^2+36 c_1{}^2\right ) \]