61.2.3 problem Problem 1(c)

Internal problem ID [15241]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 1(c)
Date solved : Friday, October 03, 2025 at 07:29:49 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+y y^{\prime }&=1 \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 61

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

\[ 2 \,2^{{2}/{3}} \int _{}^{y}\frac {1}{-2^{{2}/{3}} \textit {\_a}^{2}+4 \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 2^{{1}/{3}} c_1 \textit {\_a} +2^{{1}/{3}} \textit {\_a} \operatorname {AiryAi}\left (\textit {\_Z} \right )-2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_1 -2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right )}d \textit {\_a} -x -c_2 = 0 \]

✓ Mathematica. Time used: 57.843 (sec). Leaf size: 111

ode=D[y[x],{x,2}]+y[x]*D[y[x],x]==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2^{2/3} \left (c_2 \operatorname {AiryAiPrime}\left (\frac {x-c_1}{\sqrt [3]{2}}\right )+\operatorname {AiryBiPrime}\left (\frac {x-c_1}{\sqrt [3]{2}}\right )\right )}{c_2 \operatorname {AiryAi}\left (\frac {x-c_1}{\sqrt [3]{2}}\right )+\operatorname {AiryBi}\left (\frac {x-c_1}{\sqrt [3]{2}}\right )}\\ y(x)&\to \frac {2^{2/3} \operatorname {AiryAiPrime}\left (\frac {x-c_1}{\sqrt [3]{2}}\right )}{\operatorname {AiryAi}\left (\frac {x-c_1}{\sqrt [3]{2}}\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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