29.3.7 problem 61

Internal problem ID [4669]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 3
Problem number : 61
Date solved : Tuesday, March 04, 2025 at 07:01:55 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=1+a \left (x -y\right ) y \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 73

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

\[ y \left (x \right ) = \frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) a x +2 a^{{3}/{2}} c_{1} x +2 \sqrt {a}\, {\mathrm e}^{-\frac {a \,x^{2}}{2}}}{a \left (\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+2 c_{1} \sqrt {a}\right )} \]

✓ Mathematica. Time used: 0.567 (sec). Leaf size: 134

ode=D[y[x],x]==1+a*(x-y[x])*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\sqrt {2 \pi } c_1 x \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )+\frac {2 \left (a x+c_1 e^{-\frac {a x^2}{2}}\right )}{\sqrt {a}}}{2 \sqrt {a}+\sqrt {2 \pi } c_1 \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )} \\ y(x)\to \frac {\sqrt {\frac {2}{\pi }} e^{-\frac {a x^2}{2}}}{\sqrt {a} \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )}+x \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(x - y(x))*y(x) + Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*x*y(x) + a*y(x)**2 + Derivative(y(x), x) - 1 cannot be solved by the factorable group method