23.2.25 problem 26

Internal problem ID [5380]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 26
Date solved : Tuesday, September 30, 2025 at 12:38:26 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 28

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

\begin{align*} y &= \frac {a x -1}{a} \\ y &= \frac {\left (x -c_1 \right )^{2} a}{4}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 84

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

\begin{align*} y(x)&\to \frac {1}{4} a \left (x^2-2 \sqrt {2} c_1 x+2 c_1{}^2\right )-\frac {1}{a}+x\\ y(x)&\to \frac {1}{4} a \left (x^2+2 \sqrt {2} c_1 x+2 c_1{}^2\right )-\frac {1}{a}+x\\ y(x)&\to x-\frac {1}{a} \end{align*}

✓ Sympy. Time used: 1.117 (sec). Leaf size: 31

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

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