Internal problem ID [7725]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 145.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Abel]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}+a y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 117
dsolve(x^2*diff(y(x),x) + a*y(x)^3 - a*x^2*y(x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {1}{a x +\left (-2 a \right )^{\frac {2}{3}} \RootOf \left (\AiryBi \left (\frac {\textit {\_Z}^{2} \left (-2 a \right )^{\frac {1}{3}} x -1}{\left (-2 a \right )^{\frac {1}{3}} x}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \AiryAi \left (\frac {\textit {\_Z}^{2} \left (-2 a \right )^{\frac {1}{3}} x -1}{\left (-2 a \right )^{\frac {1}{3}} x}\right )+\AiryBi \left (1, \frac {\textit {\_Z}^{2} \left (-2 a \right )^{\frac {1}{3}} x -1}{\left (-2 a \right )^{\frac {1}{3}} x}\right ) c_{1}+\AiryAi \left (1, \frac {\textit {\_Z}^{2} \left (-2 a \right )^{\frac {1}{3}} x -1}{\left (-2 a \right )^{\frac {1}{3}} x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.483 (sec). Leaf size: 267
DSolve[x^2*y'[x] + a*y[x]^3 - a*x^2*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \operatorname {Solve}\left [\frac {\left (-\frac {1}{2^{2/3} a^{2/3} y(x)}-\frac {\sqrt [3]{a} x}{2^{2/3}}\right ) \operatorname {Ai}\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )+\operatorname {Ai}'\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )}{\left (-\frac {1}{2^{2/3} a^{2/3} y(x)}-\frac {\sqrt [3]{a} x}{2^{2/3}}\right ) \operatorname {Bi}\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )+\operatorname {Bi}'\left (\left (-\frac {\sqrt [3]{a} x}{2^{2/3}}-\frac {1}{2^{2/3} a^{2/3} y(x)}\right )^2+\frac {1}{\sqrt [3]{2} \sqrt [3]{a} x}\right )}+c_1=0,y(x)\right ] \]