60.2.74 problem 650

Internal problem ID [10661]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 650
Date solved : Monday, January 27, 2025 at 09:22:26 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.197 (sec). Leaf size: 35

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

\[ c_{1} +\frac {a^{2}}{4}+\frac {x^{2}}{4}+\frac {1}{16}-\frac {\sqrt {a^{2}+2 a x +x^{2}+4 y}}{4} = 0 \]

✓ Solution by Mathematica

Time used: 0.644 (sec). Leaf size: 39

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

\[ y(x)\to \frac {1}{4} \left (-a^2-2 a x+x^4-(1+4 c_1) x^2+4 c_1{}^2\right ) \]