60.2.72 problem 648

Internal problem ID [10659]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 648
Date solved : Tuesday, January 28, 2025 at 05:02:47 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.154 (sec). Leaf size: 49

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

\[ -\sqrt {a \,x^{4}+8 y}-4 \sqrt {a}\, \ln \left (x +1\right )+\frac {2 \left (2 x^{3}-3 x^{2}+6 x \right ) \sqrt {a}}{3}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 4.313 (sec). Leaf size: 53

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

\[ y(x)\to -\frac {a x^4}{8}+2 a \left (\int \frac {x^3}{x+1} \, dx\right )^2-4 a c_1 \int \frac {x^3}{x+1} \, dx+2 a c_1{}^2 \]