89.33.8 problem 9

Internal problem ID [24992]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 9
Date solved : Thursday, October 02, 2025 at 11:46:02 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x y^{\prime \prime }&=y^{\prime }+x^{5} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 16

ode:=x*diff(diff(y(x),x),x) = diff(y(x),x)+x^5; 
ic:=[y(1) = 1/2, D(y)(1) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {1}{24} x^{6}+\frac {3}{8} x^{2}+\frac {1}{12} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 19

ode=x*D[y[x],{x,2}]==D[y[x],x]+x^5; 
ic={y[1]==1/2,Derivative[1][y][1] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{24} \left (x^6+9 x^2+2\right ) \end{align*}

✓ Sympy. Time used: 0.193 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**5 + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0) 
ics = {y(1): 1/2, Subs(Derivative(y(x), x), x, 1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{6}}{24} + \frac {3 x^{2}}{8} + \frac {1}{12} \]