60.7.185 problem 1776 (book 6.185)

Internal problem ID [11774]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1776 (book 6.185)
Date solved : Tuesday, January 28, 2025 at 06:11:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x \left (1+x \right )^{2} y y^{\prime \prime }-x \left (1+x \right )^{2} {y^{\prime }}^{2}+2 \left (1+x \right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 31

dsolve(x*(x+1)^2*y(x)*diff(diff(y(x),x),x)-x*(x+1)^2*diff(y(x),x)^2+2*(x+1)^2*y(x)*diff(y(x),x)-a*(x+2)*y(x)^2=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {\left (x +1\right )^{a} {\mathrm e}^{\frac {c_{2} +\left (-x -1\right ) a}{x}}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 10.824 (sec). Leaf size: 47

DSolve[-(a*(2 + x)*y[x]^2) + 2*(1 + x)^2*y[x]*D[y[x],x] - x*(1 + x)^2*D[y[x],x]^2 + x*(1 + x)^2*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {c_1+\int _1^{K[2]}\frac {a K[1] (K[1]+2)}{(K[1]+1)^2}dK[1]}{K[2]^2}dK[2]\right ) \]