Internal problem ID [8383]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 46.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
\[ \boxed {y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y=x^{-2 a}-a \,x^{-a -1}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 2084
dsolve(diff(y(x),x) - x^a*y(x)^3 + 3*y(x)^2 - x^(-a)*y(x) -x^(-2*a) + a*x^(-a-1)=0,y(x), singsol=all)
\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}
✓ Solution by Mathematica
Time used: 13.471 (sec). Leaf size: 231
DSolve[y'[x] - x^a*y[x]^3 + 3*y[x]^2 - x^(-a)*y[x] -x^(-2*a) + a*x^(-a-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^{-a}-\frac {e^{\frac {2 x^{1-a}}{a-1}}}{\sqrt {-\frac {2^{\frac {3 a+1}{a-1}} x^{a+1} \left (\frac {x^{1-a}}{1-a}\right )^{\frac {a+1}{a-1}} \Gamma \left (\frac {a+1}{1-a},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a}+\frac {e^{\frac {2 x^{1-a}}{a-1}}}{\sqrt {-\frac {2^{\frac {3 a+1}{a-1}} x^{a+1} \left (\frac {x^{1-a}}{1-a}\right )^{\frac {a+1}{a-1}} \Gamma \left (\frac {a+1}{1-a},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a} \\ \end{align*}