60.2.81 problem 657

Internal problem ID [10668]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 657
Date solved : Monday, January 27, 2025 at 09:23:22 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.165 (sec). Leaf size: 26

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

\[ c_{1} +\frac {4 x^{3}}{3}-\sqrt {x^{2}+8 y-2 x +1} = 0 \]

✓ Solution by Mathematica

Time used: 0.851 (sec). Leaf size: 37

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

\[ y(x)\to \frac {1}{72} \left (16 x^6-96 c_1 x^3-9 x^2+18 x-9+144 c_1{}^2\right ) \]