62.36.5 problem Ex 5

Internal problem ID [13003]
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
Date solved : Tuesday, January 28, 2025 at 08:24:45 PM
CAS classification : [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.058 (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 &= 0 \\ y &= \frac {\sqrt {-x \left (c_{1} x^{2}+2 c_{3} x -2 c_{2} \right )}}{x^{2}} \\ y &= -\frac {\sqrt {-x \left (c_{1} x^{2}+2 c_{3} x -2 c_{2} \right )}}{x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.223 (sec). Leaf size: 60

DSolve[2*x^3*y[x]*D[y[x],{x,3}]+6*x^3*D[y[x],x]*D[y[x],{x,2}]+18*x^2*y[x]*D[y[x],{x,2}]+18*x^2*D[y[x],x]^2+36*x*y[x]*D[y[x],x]+6*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \\ y(x)\to \frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \\ \end{align*}