15.19.8 problem 8

Internal problem ID [3292]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 8
Date solved : Monday, January 27, 2025 at 07:31:28 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.082 (sec). Leaf size: 107

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

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

✓ Solution by Mathematica

Time used: 5.124 (sec). Leaf size: 171

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

\begin{align*} y(x)\to -\frac {\sqrt {-2 x^2-4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-2 x^2-4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-2 x^2+4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-2 x^2+4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}