68.15.42 problem 42

Internal problem ID [17768]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 42
Date solved : Thursday, October 02, 2025 at 02:27:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+27 x y^{\prime }+10 y&=\frac {1}{x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 24

ode:=9*x^2*diff(diff(y(x),x),x)+27*x*diff(y(x),x)+10*y(x) = 1/x; 
ic:=[y(1) = 0, D(y)(1) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {-3 \sin \left (\frac {\ln \left (x \right )}{3}\right )-\cos \left (\frac {\ln \left (x \right )}{3}\right )+1}{x} \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 81

ode=9*x^2*D[y[x],{x,2}]+27*x*D[y[x],x]+10*y[x]==1/x; 
ic={y[1]==0,Derivative[1][y][1]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sin \left (\frac {\log (x)}{3}\right ) \int _1^x\frac {\cos \left (\frac {1}{3} \log (K[1])\right )}{3 K[1]}dK[1]+\cos \left (\frac {\log (x)}{3}\right ) \int _1^x-\frac {\sin \left (\frac {1}{3} \log (K[2])\right )}{3 K[2]}dK[2]-3 \sin \left (\frac {\log (x)}{3}\right )}{x} \end{align*}

✓ Sympy. Time used: 0.312 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + 27*x*Derivative(y(x), x) + 10*y(x) - 1/x,0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {- 3 \sin {\left (\frac {\log {\left (x \right )}}{3} \right )} - \cos {\left (\frac {\log {\left (x \right )}}{3} \right )} + 1}{x} \]