76.18.7 problem Ex. 8

Internal problem ID [20114]
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. Examples on chapter III. page 38
Problem number : Ex. 8
Date solved : Sunday, October 12, 2025 at 05:34:13 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} \left (x y^{\prime }-y\right )^{2}&=a \left (1+{y^{\prime }}^{2}\right ) \left (x^{2}+y^{2}\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 7.351 (sec). Leaf size: 98

ode:=(-y(x)+x*diff(y(x),x))^2 = a*(1+diff(y(x),x)^2)*(x^2+y(x)^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x \\ y &= i x \\ y &= x \cot \left (\operatorname {RootOf}\left (-4 \textit {\_Z} -2 \int _{}^{x^{2} \csc \left (\textit {\_Z} \right )^{2}}\frac {\sqrt {-\textit {\_a}^{{17}/{2}} \left (\sqrt {\textit {\_a}}\, a -1\right ) \left (2 \sqrt {\textit {\_a}}\, a +\cos \left (2\right )-1\right )^{2} a}}{\textit {\_a}^{5} \left (2 \textit {\_a} \,a^{2}-3 \sqrt {\textit {\_a}}\, a +1+\sqrt {\textit {\_a}}\, a \cos \left (2\right )-\cos \left (2\right )\right )}d \textit {\_a} +4 c_1 \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 65.475 (sec). Leaf size: 305

ode=(x*D[y[x],x]-y[x])^2==a*(1+D[y[x],x]^2)*(x^2+y[x]^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(x**2 + y(x)**2)**(3/2)*(Derivative(y(x), x)**2 + 1) + (x*Derivative(y(x), x) - y(x))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out