23.2.5 problem 5

Internal problem ID [5360]
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 : 5
Date solved : Tuesday, September 30, 2025 at 12:36:09 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} {y^{\prime }}^{2}+x^{2}&=4 y \end{align*}

✓ Maple. Time used: 0.470 (sec). Leaf size: 136

ode:=diff(y(x),x)^2+x^2 = 4*y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_1}{2}}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_1}{2}}}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_1}{2}}}{2}\right )^{2}} \\ y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (-\frac {x \sqrt {2}\, c_1}{2}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {x \sqrt {2}\, c_1}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (-\frac {x \sqrt {2}\, c_1}{2}\right )^{2}} \\ y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, c_1}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, c_1}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, c_1}{2}\right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 1.54 (sec). Leaf size: 162

ode=(D[y[x],x])^2+x^2==4*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )+\frac {x \left (-\sqrt {4 y(x)-x^2}\right )+\left (x^2-2 y(x)\right ) \log \left (2 y(x)-x^2\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {x \sqrt {4 y(x)-x^2}+\left (x^2-2 y(x)\right ) \log \left (2 y(x)-x^2\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}-\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )=c_1,y(x)\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2 - 4*y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-x**2 + 4*y(x)) + Derivative(y(x), x) cannot be solved by