83.34.2 problem 2

Internal problem ID [19415]
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 (H) at page 118
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 01:38:50 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}&=3 y^{\prime } y \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 39

dsolve(x*y(x)*diff(y(x),x$2)+x*diff(y(x),x)^2=3*y(x)*diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {2 c_{1} x^{4}+8 c_{2}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2 c_{1} x^{4}+8 c_{2}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.274 (sec). Leaf size: 20

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

\[ y(x)\to c_2 \sqrt {x^4+2 c_1} \]