Internal problem ID [8382]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 45.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
\[ \boxed {y^{\prime }+2 \left (a^{2} x^{3}-x \,b^{2}\right ) y^{3}+3 b y^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 113
dsolve(diff(y(x),x) + 2*(a^2*x^3 - b^2*x)*y(x)^3 + 3*b*y(x)^2=0,y(x), singsol=all)
\[ c_{1} +\frac {\left (\frac {a^{2} y \left (x \right )^{2} x^{4}-y \left (x \right )^{2} b^{2} x^{2}+2 b x y \left (x \right )-1}{\left (b x y \left (x \right )-1\right )^{2}}\right )^{\frac {1}{4}} a x}{\sqrt {\frac {a \,x^{2} y \left (x \right )}{b x y \left (x \right )-1}}\, b \left (b x y \left (x \right )-1\right )}-\left (\int _{}^{\frac {a \,x^{2} y \left (x \right )}{b x y \left (x \right )-1}}\frac {\left (\textit {\_a}^{2}-1\right )^{\frac {1}{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) = 0 \]
✓ Solution by Mathematica
Time used: 0.442 (sec). Leaf size: 133
DSolve[y'[x] + 2*(a^2*x^3 - b^2*x)*y[x]^3 + 3*b*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\sqrt [4]{\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right )^2-1} \left (-\frac {\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right )^2\right )}{2 \sqrt [4]{1-\left (\frac {b}{a x}-\frac {1}{a x^2 y(x)}\right )^2}}-\frac {a x}{b}\right ),y(x)\right ] \]