60.2.102 problem 678

Internal problem ID [10689]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 678
Date solved : Tuesday, January 28, 2025 at 05:05:51 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.277 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 4.257 (sec). Leaf size: 49

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

\[ y(x)\to \frac {1}{6} \left (x^3-9 \left (\int \frac {x^3}{x+1} \, dx\right )^2+18 c_1 \int \frac {x^3}{x+1} \, dx-9 c_1{}^2\right ) \]