23.1.101 problem 100 (a)

Internal problem ID [4708]
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 : 100 (a)
Date solved : Tuesday, September 30, 2025 at 08:15:51 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=a +b y+\sqrt {A +B y} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

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

\[ x -\int _{}^{y}\frac {1}{a +b \textit {\_a} +\sqrt {B \textit {\_a} +A}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.451 (sec). Leaf size: 172

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

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

✓ Sympy. Time used: 2.479 (sec). Leaf size: 204

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

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