60.1.46 problem 46

Internal problem ID [10060]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 46
Date solved : Monday, January 27, 2025 at 06:20:06 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: 0.003 (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: 12.729 (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*}