83.23.13 problem 13

Internal problem ID [19294]
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 : 13
Date solved : Tuesday, January 28, 2025 at 01:24:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 67

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

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

✓ Solution by Mathematica

Time used: 5.413 (sec). Leaf size: 64

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

\begin{align*} y(x)\to -\sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to \sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to 0 \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}