60.7.170 problem 1761 (book 6.170)

Internal problem ID [11759]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1761 (book 6.170)
Date solved : Tuesday, January 28, 2025 at 06:11:17 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 112

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

\begin{align*} y &= \frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{-a +1} \left (\int x^{a -1} fd x \right )+x^{-a +1} c_{1} -\int fd x -c_{2} \right )}}{a -1} \\ y &= -\frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{-a +1} \left (\int x^{a -1} fd x \right )+x^{-a +1} c_{1} -\int fd x -c_{2} \right )}}{a -1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 26.209 (sec). Leaf size: 108

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

\begin{align*} y(x)\to -\sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \\ y(x)\to \sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \\ \end{align*}