60.1.426 problem 437

Internal problem ID [10440]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 437
Date solved : Sunday, March 30, 2025 at 04:43:34 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Clairaut]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 36

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

\begin{align*} y &= -\frac {a}{4 x} \\ y &= c_1 x -\sqrt {c_1 a} \\ y &= c_1 x +\sqrt {c_1 a} \\ \end{align*}

✓ Mathematica. Time used: 0.312 (sec). Leaf size: 64

ode=y[x]^2 - (a + 2*x*y[x])*D[y[x],x] + x^2*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 5.852 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x)**2 - (a + 2*x*y(x))*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \begin {cases} \frac {C_{1} x^{2} - a + \sqrt {a^{2}}}{4 x} & \text {for}\: a = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1} \left (C_{1} x - 2 a\right )}{4 a} & \text {for}\: a \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1} x^{2} - a - \sqrt {a^{2}}}{4 x} & \text {for}\: a = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1} \left (C_{1} x - 2 a\right )}{4 a} & \text {for}\: a \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]