2.7.16 problem 16

Internal problem ID [822]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.1, second order linear equations. Page 299
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 04:15:48 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 15

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0; 
ic:=[y(1) = 2, D(y)(1) = 3]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 3 \sin \left (\ln \left (x \right )\right )+2 \cos \left (\ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 16

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0; 
ic={y[1]==2,Derivative[1][y][1]==3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 3 \sin (\log (x))+2 \cos (\log (x)) \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + y(x),0) 
ics = {y(1): 2, Subs(Derivative(y(x), x), x, 1): 3} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 3 \sin {\left (\log {\left (x \right )} \right )} + 2 \cos {\left (\log {\left (x \right )} \right )} \]