60.2.87 problem 663

Internal problem ID [10674]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 663
Date solved : Tuesday, January 28, 2025 at 05:03:31 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {2 a +x^{2} \sqrt {-y^{2}+4 a x}}{y} \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 5.535 (sec). Leaf size: 159

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

\begin{align*} y(x)\to -\frac {\sqrt {a^6 x \left (36 a-x^5\right )+2 a^3 e^{c_1} x^3-e^{2 c_1}}}{3 a^3} \\ y(x)\to \frac {\sqrt {a^6 x \left (36 a-x^5\right )+2 a^3 e^{c_1} x^3-e^{2 c_1}}}{3 a^3} \\ y(x)\to -\frac {\sqrt {a^6 x \left (36 a-x^5\right )}}{3 a^3} \\ y(x)\to \frac {\sqrt {a^6 x \left (36 a-x^5\right )}}{3 a^3} \\ \end{align*}