53.4.8 problem 9

Internal problem ID [8496]
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 : 9
Date solved : Wednesday, March 05, 2025 at 06:01:15 AM
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: 0.009 (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,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.034 (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]
 

\[ y(x)\to \frac {1}{24} \left (x^6+9 x^2+2\right ) \]

✓ Sympy. Time used: 0.278 (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)
 

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