Internal problem ID [10325]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {x \left (x +2 y\right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (x +y\right ) y^{\prime }+2 y+x^{2}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 75
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 \left (x \right ) = \frac {-3 x^{2}+\sqrt {-3 x^{5}+9 x^{4}-36 x^{2} c_{1} +36 c_{2} x}}{6 x} \\ y \left (x \right ) = -\frac {3 x^{2}+\sqrt {-3 x^{5}+9 x^{4}-36 x^{2} c_{1} +36 c_{2} x}}{6 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 100
DSolve[x*(x+2*y[x])*y''[x]+2*x*(y'[x])^2+4*(x+y[x])*y'[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 (-\left ((x-3) x^3\right )+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 (-\left ((x-3) x^3\right )+12 c_2 x+12 c_1\right )}\right ) \\ \end{align*}