23.2.331 problem 373

Internal problem ID [5686]
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 : 373
Date solved : Tuesday, September 30, 2025 at 01:57:44 PM
CAS classification : [[_homogeneous, `class G`], _Clairaut]

\begin{align*} 2 \sqrt {a y^{\prime }}+x y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.203 (sec). Leaf size: 15

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

\[ y = 2 \sqrt {a c_1}+c_1 x \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to 2 \sqrt {a c_1}+c_1 x\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 3.656 (sec). Leaf size: 41

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

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