60.1.441 problem 444

Internal problem ID [10455]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 444
Date solved : Monday, January 27, 2025 at 07:45:58 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 1.269 (sec). Leaf size: 124

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

\begin{align*} y &= 4 x \\ y &= 0 \\ y &= -\frac {2 c_{1}^{2} \left (-c_{1} \sqrt {2}+x \right )}{-2 c_{1}^{2}+x^{2}} \\ y &= -\frac {2 c_{1}^{2} \left (c_{1} \sqrt {2}+x \right )}{-2 c_{1}^{2}+x^{2}} \\ y &= \frac {c_{1}^{3} \sqrt {2}-2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ y &= \frac {c_{1}^{2} \left (c_{1} \sqrt {2}+2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.275 (sec). Leaf size: 61

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

\begin{align*} y(x)\to -\frac {4 e^{-2 c_1}}{-2+e^{2 c_1} x} \\ y(x)\to -\frac {4 e^{-2 c_1}}{2+e^{2 c_1} x} \\ y(x)\to 0 \\ y(x)\to 4 x \\ \end{align*}