60.2.73 problem 649

Internal problem ID [10660]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 649
Date solved : Monday, January 27, 2025 at 09:22:22 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.162 (sec). Leaf size: 27

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

\[ c_{1} +\frac {x^{2}}{8}+\frac {17}{128}-\frac {\sqrt {x^{2}+8 y-2 x +1}}{16} = 0 \]

✓ Solution by Mathematica

Time used: 0.786 (sec). Leaf size: 36

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

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