60.2.89 problem 665

Internal problem ID [10676]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 665
Date solved : Tuesday, January 28, 2025 at 05:04:08 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.145 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.693 (sec). Leaf size: 39

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

\[ y(x)\to -\frac {a x^4}{8}+2 a \log ^2(x+1)-4 a c_1 \log (x+1)+2 a c_1{}^2 \]