83.47.13 problem Ex 13 page 91

Internal problem ID [19590]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VI. Homogeneous linear equations with variable coefficients
Problem number : Ex 13 page 91
Date solved : Tuesday, January 28, 2025 at 08:32:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 x^{2} y y^{\prime \prime }+4 y^{2}&=x^{2} {y^{\prime }}^{2}+2 x y y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

Time used: 0.443 (sec). Leaf size: 20

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

\[ y(x)\to c_2 x^2 (\log (x)-2 c_1){}^2 \]