60.2.65 problem 641

Internal problem ID [10652]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 641
Date solved : Tuesday, January 28, 2025 at 05:01:57 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.279 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.340 (sec). Leaf size: 33

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

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