Internal problem ID [10313]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60.
Exact equation. Integrating factor. Page 139
Problem number: Ex 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 56
dsolve(2*x^3*y(x)*diff(y(x),x$3)+6*x^3*diff(y(x),x)*diff(y(x),x$2)+18*x^2*y(x)*diff(y(x),x$2)+18*x^2*diff(y(x),x)^2+36*x*y(x)*diff(y(x),x)+6*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {\sqrt {-x \left (x^{2} c_{1}+2 c_{2} x -2 c_{3}\right )}}{x^{2}} \\ y \relax (x ) = -\frac {\sqrt {-x \left (x^{2} c_{1}+2 c_{2} x -2 c_{3}\right )}}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.194 (sec). Leaf size: 58
DSolve[2*x^3*y[x]*y'''[x]+6*x^3*y'[x]*y''[x]+18*x^2*y[x]*y''[x]+18*x^2*y'[x]^2+36*x*y[x]*y'[x]+6*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x (c_1 x+c_3)+2 c_2}}{x^{3/2}} \\ y(x)\to \frac {\sqrt {x (c_1 x+c_3)+2 c_2}}{x^{3/2}} \\ \end{align*}