23.2.53 problem 54

Internal problem ID [5408]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 54
Date solved : Tuesday, September 30, 2025 at 12:39:58 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} {y^{\prime }}^{2}+\left (b x +a \right ) y^{\prime }+c&=b y \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 50

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

\begin{align*} y &= \frac {-b^{2} x^{2}-2 a b x -a^{2}+4 c}{4 b} \\ y &= \frac {c_1^{2}+\left (b x +a \right ) c_1 +c}{b} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \frac {c+c_1 (a+b x+c_1)}{b}\\ y(x)&\to -\frac {a^2+2 a b x+b^2 x^2-4 c}{4 b} \end{align*}

✓ Sympy. Time used: 2.027 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {- a^{2} + b^{2} \left (C_{1} - x\right )^{2} - b x \left (2 a + b x\right ) + 4 c}{4 b} \]