67.2.34 problem Problem 13

Internal problem ID [13920]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 13
Date solved : Monday, March 31, 2025 at 08:18:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

ode:=x^2*diff(diff(y(x),x),x)-4*x^2*diff(y(x),x)+(x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, {\mathrm e}^{2 x} \left (\operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right ) c_2 +\operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 67

ode=x^2*D[y[x],{x,2}]-4*x^2*D[y[x],x]+(x^2+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{2 x} \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (\frac {i \sqrt {3}}{2},-i \sqrt {3} x\right )+c_2 \operatorname {BesselY}\left (\frac {i \sqrt {3}}{2},-i \sqrt {3} x\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None