78.2.25 problem 8.d

Internal problem ID [20977]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 8.d
Date solved : Thursday, October 02, 2025 at 07:00:55 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+3 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=3 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 32

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

\[ y = -3 x^{{3}/{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )-\cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to 3 x^{3/2} \left (\cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )-\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )\right ) \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 39

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

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