29.33.27 problem 990

Internal problem ID [5566]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 990
Date solved : Tuesday, March 04, 2025 at 10:19:21 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.353 (sec). Leaf size: 135

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

\begin{align*} y \left (x \right ) &= x -\sqrt {2}\, a \\ y \left (x \right ) &= x +\sqrt {2}\, a \\ y \left (x \right ) &= x +\operatorname {RootOf}\left (-2 x -\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} +2 c_{1} \right ) \\ y \left (x \right ) &= x +\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}-\frac {-2 a^{2}+\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} +2 c_{1} \right ) \\ \end{align*}

✓ Mathematica. Time used: 118.37 (sec). Leaf size: 35180

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

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✓ Sympy. Time used: 51.465 (sec). Leaf size: 146

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

\[ \left [ y{\left (x \right )} = C_{1} - x - \int \limits ^{- C_{2} + x} \frac {r^{2}}{- r^{2} - r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr + \int \limits ^{- C_{2} + x} \frac {r \sqrt {- r^{2} + 2 a^{2}}}{- r^{2} - r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr, \ y{\left (x \right )} = C_{1} - x - \int \limits ^{- C_{2} + x} \frac {r^{2}}{- r^{2} + r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr - \int \limits ^{- C_{2} + x} \frac {r \sqrt {- r^{2} + 2 a^{2}}}{- r^{2} + r \sqrt {- r^{2} + 2 a^{2}} + 2 a^{2}}\, dr\right ] \]