83.27.11 problem 11

Internal problem ID [19357]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (A) at page 104
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 01:35:21 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 35

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

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {2 \ln \left (x \right ) c_{1} +2 c_{2}} \\ y \left (x \right ) &= -\sqrt {2 \ln \left (x \right ) c_{1} +2 c_{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.276 (sec). Leaf size: 19

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

\[ y(x)\to c_2 \sqrt {2 \log (x)+c_1} \]