60.2.156 problem 732

Internal problem ID [10743]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 732
Date solved : Tuesday, January 28, 2025 at 05:10:21 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.302 (sec). Leaf size: 43

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

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

✓ Solution by Mathematica

Time used: 1.639 (sec). Leaf size: 56

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

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