83.49.9 problem Ex 9 page 126

Internal problem ID [19551]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VIII. Linear equations of second order
Problem number : Ex 9 page 126
Date solved : Monday, March 31, 2025 at 07:33:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 15

ode:=x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+(x^2+2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (c_1 \sin \left (x \right )+c_2 \cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 33

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

\[ y(x)\to c_1 e^{-i x} x-\frac {1}{2} i c_2 e^{i x} x \]

✓ Sympy. Time used: 0.213 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = x^{\frac {3}{2}} \left (C_{1} J_{\frac {1}{2}}\left (x\right ) + C_{2} Y_{\frac {1}{2}}\left (x\right )\right ) \]