62.38.8 problem Ex 8

Internal problem ID [13019]
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 62. Summary. Page 144
Problem number : Ex 8
Date solved : Tuesday, January 28, 2025 at 04:49:21 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x \left (x +2 y\right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (x +y\right ) y^{\prime }+2 y+x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 80

dsolve(x*(x+2*y(x))*diff(y(x),x$2)+2*x*(diff(y(x),x))^2+4*(x+y(x))*diff(y(x),x)+2*y(x)+x^2=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {-3 x^{2}+\sqrt {3}\, \sqrt {-x \left (x^{4}-3 x^{3}+12 c_{1} x -12 c_{2} \right )}}{6 x} \\ y &= \frac {-\sqrt {3}\, \sqrt {-x \left (x^{4}-3 x^{3}+12 c_{1} x -12 c_{2} \right )}-3 x^{2}}{6 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.436 (sec). Leaf size: 104

DSolve[x*(x+2*y[x])*D[y[x],{x,2}]+2*x*(D[y[x],x])^2+4*(x+y[x])*D[y[x],x]+2*y[x]+x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (-3 x-\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (-x^4+3 x^3+12 c_2 x+12 c_1\right )}\right ) \\ y(x)\to \frac {1}{6} \left (-3 x+\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (-x^4+3 x^3+12 c_2 x+12 c_1\right )}\right ) \\ \end{align*}