82.14.2 problem Ex. 2

Internal problem ID [18737]
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. 2
Date solved : Monday, March 31, 2025 at 06:05:27 PM
CAS classification : [_quadrature]

\begin{align*} y&=-a y^{\prime }+\frac {c +a \arcsin \left (y^{\prime }\right )}{\sqrt {1-{y^{\prime }}^{2}}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 69

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

\begin{align*} y &= c \\ x -\int _{}^{y}\csc \left (\operatorname {RootOf}\left (-\cos \left (\textit {\_Z} \right )^{2} \sin \left (\textit {\_Z} \right )^{2} a^{2}-2 \cos \left (\textit {\_Z} \right )^{2} \sin \left (\textit {\_Z} \right ) \textit {\_a} a -\cos \left (\textit {\_Z} \right )^{2} \textit {\_a}^{2}+a^{2} \textit {\_Z}^{2}+2 a c \textit {\_Z} +c^{2}\right )\right )d \textit {\_a} -c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.865 (sec). Leaf size: 74

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

\[ \text {Solve}\left [\left \{x=\frac {K[1] (a \arcsin (K[1])+c)}{\sqrt {1-K[1]^2}}+c_1,y(x)=\frac {a \arcsin (K[1])-a K[1] \sqrt {1-K[1]^2}+c}{\sqrt {1-K[1]^2}}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

NotImplementedError : multiple generators [_X0, asin(_X0), sqrt(1 - _X0**2)] 
No algorithms are implemented to solve equation _X0*a*sqrt(1 - _X0**2) - a*asin(_X0) - c + sqrt(1 - _X0**2)*y(x)