74.16.12 problem 12

Internal problem ID [16447]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 12
Date solved : Monday, March 31, 2025 at 02:53:35 PM
CAS classification : [_Hermite]

\begin{align*} y^{\prime \prime }-x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

Order:=6; 
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-2 x^{2}+\frac {1}{3} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode=D[y[x],{x,2}]-x*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{40}-\frac {x^3}{2}+x\right )+c_1 \left (\frac {x^4}{3}-2 x^2+1\right ) \]

✓ Sympy. Time used: 0.717 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{3} - 2 x^{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{2}\right ) + O\left (x^{6}\right ) \]