60.2.133 problem 709

Internal problem ID [10720]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 709
Date solved : Tuesday, January 28, 2025 at 05:07:52 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 3.426 (sec). Leaf size: 143

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

\begin{align*} y(x)\to -\frac {1}{6} \sqrt {144 a x-\left (2 x^3-3 x^2+6 x+6 c_1\right ){}^2+12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)-36 \log ^2(x+1)} \\ y(x)\to \frac {1}{6} \sqrt {144 a x-\left (2 x^3-3 x^2+6 x+6 c_1\right ){}^2+12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)-36 \log ^2(x+1)} \\ \end{align*}