60.7.184 problem 1775 (book 6.184)

Internal problem ID [11773]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1775 (book 6.184)
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*} a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 119

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

\begin{align*} y &= 0 \\ y &= x^{\frac {-\sqrt {\left (-4 a -4 b \right ) d +\left (-c +a \right )^{2}}+a -c}{2 a +2 b}} \left (\frac {a^{2}+\left (-2 c -4 d \right ) a -4 b d +c^{2}}{\left (a +b \right )^{2} \left (x^{\frac {\sqrt {\left (-4 a -4 b \right ) d +\left (-c +a \right )^{2}}}{a}} c_{1} -c_{2} \right )^{2}}\right )^{-\frac {a}{2 a +2 b}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 45.015 (sec). Leaf size: 92

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

\[ y(x)\to c_2 \exp \left (-\frac {\log (x) \left (a \left (\sqrt {\frac {a^2-2 a (c+2 d)-4 b d+c^2}{a^2}}-1\right )+c\right )-2 a \log \left (x^{\sqrt {\frac {a^2-2 a (c+2 d)-4 b d+c^2}{a^2}}}+c_1\right )}{2 (a+b)}\right ) \]