83.23.16 problem 16

Internal problem ID [19297]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 01:24:42 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.088 (sec). Leaf size: 129

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

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

✓ Solution by Mathematica

Time used: 0.742 (sec). Leaf size: 79

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

\begin{align*} y(x)\to -\frac {x (\cosh (2 c_1)+\sinh (2 c_1))}{x+i \cosh (c_1)+i \sinh (c_1)} \\ y(x)\to \frac {x (\cosh (2 c_1)+\sinh (2 c_1))}{-x+i \cosh (c_1)+i \sinh (c_1)} \\ y(x)\to 0 \\ \end{align*}