23.2.143 problem 146

Internal problem ID [5498]
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 : 146
Date solved : Tuesday, September 30, 2025 at 12:45:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.706 (sec). Leaf size: 44

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

\[ y = \frac {x \left (\operatorname {LambertW}\left (-c_1 \,{\mathrm e} x^{4}\right )-1\right )}{2 \operatorname {LambertW}\left (-c_1 \,{\mathrm e} x^{4}\right ) \sqrt {-\frac {1}{\operatorname {LambertW}\left (-c_1 \,{\mathrm e} x^{4}\right )}}} \]

✓ Mathematica. Time used: 1.391 (sec). Leaf size: 183

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

\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}{\left (\sqrt {\frac {y(x)}{x}-1}-\sqrt {\frac {y(x)}{x}+1}\right )^2}=\log (x)+c_1,y(x)\right ]\\ \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}{\left (\sqrt {\frac {y(x)}{x}-1}+\sqrt {\frac {y(x)}{x}+1}\right )^2}=-\log (x)+c_1,y(x)\right ] \end{align*}

✓ Sympy. Time used: 8.334 (sec). Leaf size: 61

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

\[ \left [ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {1 - u_{1}^{2}}}{u_{1} \left (\sqrt {1 - u_{1}^{2}} + 1\right )}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {1 - u_{1}^{2}}}{u_{1} \left (\sqrt {1 - u_{1}^{2}} - 1\right )}\, du_{1}}\right ] \]