23.1.249 problem 244 (a)

Internal problem ID [4856]
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 : 244 (a)
Date solved : Tuesday, September 30, 2025 at 08:45:11 AM
CAS classification : [_separable]

\begin{align*} 2 x y^{\prime }+4 y+a +\sqrt {a^{2}-4 b -4 c y}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 34

ode:=2*x*diff(y(x),x)+4*y(x)+a+(a^2-4*b-4*c*y(x))^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (x \right )+2 \int _{}^{y}\frac {1}{4 \textit {\_a} +a +\sqrt {-4 \textit {\_a} c +a^{2}-4 b}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.497 (sec). Leaf size: 177

ode=2*x*D[y[x],x]+4*y[x]+a+Sqrt[a^2-4*b- 4*c*y[x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{4} \left (\log \left (c \left (\sqrt {a^2-4 (\text {$\#$1} c+b)}+4 \text {$\#$1}+a\right )\right )-\frac {2 c \arctan \left (\frac {c-2 \sqrt {a^2-4 (\text {$\#$1} c+b)}}{\sqrt {-4 a^2-4 a c+16 b-c^2}}\right )}{\sqrt {-4 a^2-4 a c+16 b-c^2}}\right )\&\right ]\left [-\frac {\log (x)}{2}+c_1\right ]\\ y(x)&\to \frac {1}{8} \left (-\sqrt {(2 a+c)^2-16 b}-2 a-c\right )\\ y(x)&\to \frac {1}{8} \left (\sqrt {(2 a+c)^2-16 b}-2 a-c\right ) \end{align*}

✓ Sympy. Time used: 2.534 (sec). Leaf size: 192

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a + 2*x*Derivative(y(x), x) + sqrt(a**2 - 4*b - 4*c*y(x)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \begin {cases} \frac {c \operatorname {atan}{\left (\frac {- \frac {c}{2} + \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}}}{\sqrt {- a^{2} - a c + 4 b - \frac {c^{2}}{4}}} \right )}}{4 \sqrt {- a^{2} - a c + 4 b - \frac {c^{2}}{4}}} + \frac {\log {\left (a c + c \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}} + 4 c y{\left (x \right )} \right )}}{4} & \text {for}\: c \neq 0 \wedge 4 a^{2} + 4 a c - 16 b + c^{2} \neq 0 \\- \frac {c}{4 \left (- \frac {c}{2} + \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}}\right )} + \frac {\log {\left (a c + c \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}} + 4 c y{\left (x \right )} \right )}}{4} & \text {for}\: c \neq 0 \\\frac {\log {\left (a + \sqrt {a^{2} - 4 b} + 4 y{\left (x \right )} \right )}}{4} & \text {otherwise} \end {cases} = C_{1} - \frac {\log {\left (x \right )}}{2} \]