54.1.388 problem 399

Internal problem ID [11702]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 399
Date solved : Tuesday, September 30, 2025 at 10:11:36 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 22

ode:=2*diff(y(x),x)^2+(x-1)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\left (x -1\right )^{2}}{8} \\ y &= c_1 \left (2 c_1 +x -1\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 28

ode=-y[x] + (-1 + x)*D[y[x],x] + 2*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 (x-1+2 c_1)\\ y(x)&\to -\frac {1}{8} (x-1)^2 \end{align*}

✓ Sympy. Time used: 1.403 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*Derivative(y(x), x) - y(x) + 2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 C_{1}^{2} - C_{1} x + \frac {x}{4} - \frac {1}{8} \]