23.2.70 problem 72

Internal problem ID [5425]
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 : 72
Date solved : Tuesday, September 30, 2025 at 12:42:03 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}+\left (a x +b y\right ) y^{\prime }+a b x y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

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

\begin{align*} y &= -\frac {a \,x^{2}}{2}+c_1 \\ y &= c_1 \,{\mathrm e}^{-b x} \\ \end{align*}

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to c_1 e^{-b x}\\ y(x)&\to -\frac {a x^2}{2}+c_1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 19

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

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