60.2.114 problem 690

Internal problem ID [10701]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 690
Date solved : Tuesday, January 28, 2025 at 05:06:09 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=\frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4+4 x} \end{align*}

✓ Solution by Maple

Time used: 0.274 (sec). Leaf size: 40

dsolve(diff(y(x),x) = 1/4*(-x^2+1+4*x^3*(x^2-2*x+1+8*y(x))^(1/2))/(x+1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.349 (sec). Leaf size: 108

DSolve[D[y[x],x] == (1/4 - x^2/4 + x^3*Sqrt[1 - 2*x + x^2 + 8*y[x]])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {2 x^6}{9}-\frac {2 x^5}{3}+\frac {11 x^4}{6}-\frac {2}{3} (3+2 c_1) x^3+\left (\frac {15}{8}+2 c_1\right ) x^2+\left (\frac {4 x^3}{3}-2 x^2+4 x-4 c_1\right ) \log \left (\frac {1}{x+1}\right )+2 \log ^2\left (\frac {1}{x+1}\right )+\left (\frac {1}{4}-4 c_1\right ) x-\frac {1}{8}+2 c_1{}^2 \]