87.10.7 problem 8

Internal problem ID [23408]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 79
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:41:11 PM
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 }&=2 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.168 (sec). Leaf size: 56

ode:=diff(diff(y(x),x),x)+y(x)*diff(y(x),x) = 2; 
ic:=[y(0) = 0, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 38.182 (sec). Leaf size: 57

ode=D[y[x],{x,2}]+y[x]*D[y[x],x]==2; 
ic={y[0]==0,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 \left (\operatorname {AiryAiPrime}\left (-\frac {1}{2}\right ) \operatorname {AiryBiPrime}\left (x-\frac {1}{2}\right )-\operatorname {AiryBiPrime}\left (-\frac {1}{2}\right ) \operatorname {AiryAiPrime}\left (x-\frac {1}{2}\right )\right )}{\operatorname {AiryAiPrime}\left (-\frac {1}{2}\right ) \operatorname {AiryBi}\left (x-\frac {1}{2}\right )-\operatorname {AiryBiPrime}\left (-\frac {1}{2}\right ) \operatorname {AiryAi}\left (x-\frac {1}{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)) - 2,0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(2 - Derivative(y(x), (x, 2)))/y(x) + Derivative(y(x), x) canno