60.2.82 problem 658

Internal problem ID [10669]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 658
Date solved : Monday, January 27, 2025 at 09:23:26 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x^{2}-1-4 \sqrt {x^{2}-2 x +1+8 y}}{4 \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 4.967 (sec). Leaf size: 28

dsolve(diff(y(x),x) = -1/4*(x^2-1-4*(x^2-2*x+1+8*y(x))^(1/2))/(x+1),y(x), singsol=all)
 

\[ c_{1} +4 \ln \left (x +1\right )-\frac {1}{4}-\sqrt {x^{2}+8 y-2 x +1} = 0 \]

✓ Solution by Mathematica

Time used: 1.238 (sec). Leaf size: 46

DSolve[D[y[x],x] == (1/4 - x^2/4 + Sqrt[1 - 2*x + x^2 + 8*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{8} \left (-x^2+2 x-1+16 c_1{}^2\right )+2 \log ^2\left (\frac {1}{x+1}\right )+4 c_1 \log \left (\frac {1}{x+1}\right ) \]