60.2.100 problem 676

Internal problem ID [10687]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 676
Date solved : Tuesday, January 28, 2025 at 05:05:44 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.398 (sec). Leaf size: 53

dsolve(diff(y(x),x) = 1/2*(x+1+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3/(x+1),y(x), singsol=all)
 

\[ \frac {3 x^{5}-4 x^{4}+6 x^{3}+12 \ln \left (x +1\right ) x +6 c_{1} x -12 x^{2}-6 \sqrt {4 x^{2} y+1}}{6 x} = 0 \]

✓ Solution by Mathematica

Time used: 1.462 (sec). Leaf size: 120

DSolve[D[y[x],x] == (1/2 + x/2 + x^6*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {x^8}{16}-\frac {x^7}{6}+\frac {13 x^6}{36}-\frac {5 x^5}{6}-\frac {1}{12} (-11+6 c_1) x^4+\left (-1+\frac {2 c_1}{3}\right ) x^3-\frac {1}{4 x^2}-(-1+c_1) x^2+\left (\frac {x^4}{2}-\frac {2 x^3}{3}+x^2-2 x-2 c_1\right ) \log (x+1)+\log ^2(x+1)+2 c_1 x+c_1{}^2 \]