60.2.289 problem 866

Internal problem ID [10876]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 866
Date solved : Tuesday, January 28, 2025 at 05:27:46 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x}{2}-\frac {a}{2}+\sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y} \end{align*}

✓ Solution by Maple

Time used: 0.270 (sec). Leaf size: 37

dsolve(diff(y(x),x) = -1/2*x-1/2*a+(x^2+2*a*x+a^2+4*y(x))^(1/2)+x^2*(x^2+2*a*x+a^2+4*y(x))^(1/2)+x^3*(x^2+2*a*x+a^2+4*y(x))^(1/2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.884 (sec). Leaf size: 85

DSolve[D[y[x],x] == -1/2*a - x/2 + Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]] + x^2*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]] + x^3*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {a^2}{4}-\frac {a x}{2}+\frac {x^8}{16}+\frac {x^7}{6}+\frac {x^6}{9}+\frac {x^5}{2}-\frac {1}{6} (-4+3 c_1) x^4-\frac {2 c_1 x^3}{3}+\frac {3 x^2}{4}-2 c_1 x+c_1{}^2 \]