Internal problem ID [8421]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 841.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{\frac {3}{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{\frac {3}{2}} b \,x^{2}+a^{\frac {5}{2}} y^{4}}{a \,x^{2} y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 97
dsolve(diff(y(x),x) = (b*x^3+c^2*a^(1/2)-2*c*b*x^2*a^(1/2)+2*c*y(x)^2*a^(3/2)+b^2*x^4*a^(1/2)-2*y(x)^2*a^(3/2)*b*x^2+a^(5/2)*y(x)^4)/a/x^2/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {2 \sqrt {a^{\frac {3}{2}} \left (x c_{1}+1\right ) \left (\left (x c_{1}+1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right )}}{a^{\frac {3}{2}} \left (2 x c_{1}+2\right )} \\ y \relax (x ) = \frac {\sqrt {a^{\frac {3}{2}} \left (x c_{1}+1\right ) \left (\left (x c_{1}+1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right )}}{a^{\frac {3}{2}} \left (x c_{1}+1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 9.074 (sec). Leaf size: 390
DSolve[y'[x] == (Sqrt[a]*c^2 - 2*Sqrt[a]*b*c*x^2 + b*x^3 + Sqrt[a]*b^2*x^4 + 2*a^(3/2)*c*y[x]^2 - 2*a^(3/2)*b*x^2*y[x]^2 + a^(5/2)*y[x]^4)/(a*x^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-2 a^{5/2} \left (c-b x^2\right )+4 a^3 b x \left (b x^2-c\right )+a^2 x+4 \sqrt {a} b c_1 \left (b x^2-c\right )+2 b c_1 x}}{\sqrt {2} \sqrt {2 a^{3/2} b c_1+a^{7/2}+2 a^4 b x}} \\ y(x)\to \frac {\sqrt {-2 a^{5/2} \left (c-b x^2\right )+4 a^3 b x \left (b x^2-c\right )+a^2 x+4 \sqrt {a} b c_1 \left (b x^2-c\right )+2 b c_1 x}}{\sqrt {2} \sqrt {2 a^{3/2} b c_1+a^{7/2}+2 a^4 b x}} \\ y(x)\to -\frac {\sqrt {-b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {-a^{3/2} b}} \\ y(x)\to \frac {\sqrt {-b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {-a^{3/2} b}} \\ y(x)\to -\frac {\sqrt {b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {a^{3/2} b}} \\ y(x)\to \frac {\sqrt {b \left (x-2 \sqrt {a} \left (c-b x^2\right )\right )}}{\sqrt {2} \sqrt {a^{3/2} b}} \\ \end{align*}