76.14.3 problem Ex. 3

Internal problem ID [20094]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 33
Problem number : Ex. 3
Date solved : Thursday, October 02, 2025 at 05:25:40 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.408 (sec). Leaf size: 136

ode:=4*y(x) = x^2+diff(y(x),x)^2; 
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 {\sqrt {2}\, x c_1}{2}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_1}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_1}{2}\right )^{2}} \\ y &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_1}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_1}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_1}{2}\right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 1.358 (sec). Leaf size: 162

ode=4*y[x]==x^2+D[y[x],x]^2; 
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