60.2.136 problem 712

Internal problem ID [10723]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 712
Date solved : Tuesday, January 28, 2025 at 05:08:40 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.283 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 1.324 (sec). Leaf size: 49

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

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