60.7.186 problem 1777 (book 6.186)
Internal
problem
ID
[11775]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1777
(book
6.186)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:22 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} 8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.104 (sec). Leaf size: 53
dsolve(8*(-x^3+1)*y(x)*diff(diff(y(x),x),x)-4*(-x^3+1)*diff(y(x),x)^2-12*x^2*y(x)*diff(y(x),x)+3*x*y(x)^2=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= \frac {x {\left (\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right ) c_{1} +\frac {c_{2} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right )}{2}\right )}^{2}}{c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 61.824 (sec). Leaf size: 540
DSolve[3*x*y[x]^2 - 12*x^2*y[x]*D[y[x],x] - 4*(1 - x^3)*D[y[x],x]^2 + 8*(1 - x^3)*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_2 \exp \left (\int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right ) \left (8 \left (K[3]^3-1\right )+\exp \left (2 \int _1^{K[3]}\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right ) c_1 \left (2 K[3]^2+2 K[3]+\sqrt {2 K[3]+i \sqrt {3}+1} \sqrt {6 K[3]-3 i \sqrt {3}+3}+2\right )+\exp \left (2 \int _1^{K[3]}\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right ) \left (2 K[3]^2+2 K[3]+\sqrt {2 K[3]+i \sqrt {3}+1} \sqrt {6 K[3]-3 i \sqrt {3}+3}+2\right ) \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right )dK[2]\right )}{4 \left (K[3]^3-1\right ) \left (c_1+\int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {8 K[1]^2+2 K[1]+\sqrt {2 K[1]+i \sqrt {3}+1} \sqrt {6 K[1]-3 i \sqrt {3}+3}+2}{8 \left (K[1]^3-1\right )}dK[1]\right )dK[2]\right )}dK[3]\right )
\]