78.3.10 problem 2.e

Internal problem ID [21006]
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 4, Series solutions. Problems section 4.9
Problem number : 2.e
Date solved : Thursday, October 02, 2025 at 07:01:20 PM
CAS classification : [_Hermite]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 22

Order:=5; 
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)+3*y(x) = 0; 
ic:=[y(0) = 2]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.256 (sec). Leaf size: 39

ode=D[y[x],{x,2}]-x*D[y[x],x]+3*y[x]==0; 
ic={y[0]==2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
 

\[ y(x)\to \frac {x^4}{4}+2 \sqrt {2} c_1 x^3-3 x^2-6 \sqrt {2} c_1 x+2 \]

✓ Sympy. Time used: 0.199 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + 3*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{8} - \frac {3 x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{3}\right ) + O\left (x^{6}\right ) \]