33.1.3 problem problem 46

Internal problem ID [6021]
Book : Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section : Abel ODE with constant invariant
Problem number : problem 46
Date solved : Monday, January 27, 2025 at 01:32:49 PM
CAS classification : [_Abel]

\begin{align*} y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (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: 12.748 (sec). Leaf size: 231

DSolve[D[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],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*}