53.4.9 problem 10

Internal problem ID [8497]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 06:01:17 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+x&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=-1\\ y^{\prime }\left (2\right )&=-{\frac {1}{2}} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 16

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

\[ y = -\frac {x^{2}}{4}+\ln \left (x \right )-\ln \left (2\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 19

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

\[ y(x)\to \log \left (\frac {x}{2}\right )-\frac {x^2}{4} \]

✓ Sympy. Time used: 0.181 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = - \frac {x^{2}}{4} + \log {\left (x \right )} - \log {\left (2 \right )} \]