23.1.205 problem 201 (b)

Internal problem ID [4812]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 201 (b)
Date solved : Tuesday, September 30, 2025 at 08:42:09 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 40

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

\[ x^{-a -1} y+x^{-a -1} \sqrt {y^{2}-b^{2} x^{2}}-c_1 = 0 \]

✓ Mathematica. Time used: 0.468 (sec). Leaf size: 31

ode=x*D[y[x],x]==y[x]+a*Sqrt[y[x]^2-b^2*x^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -b x \cos (i a \log (x)+c_1)\\ y(x)&\to b x \text {Interval}[\{-1,1\}] \end{align*}

✓ Sympy. Time used: 0.694 (sec). Leaf size: 39

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

\[ \log {\left (x \right )} = C_{1} + \begin {cases} \frac {\operatorname {acosh}{\left (\frac {y{\left (x \right )}}{b x} \right )}}{a} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{b^{2} x^{2}}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{b x} \right )}}{a} & \text {otherwise} \end {cases} \]