78.3.5 problem 1.e

Internal problem ID [21001]
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 : 1.e
Date solved : Thursday, October 02, 2025 at 07:01:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

Order:=5; 
ode:=diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 17

ode=D[y[x],{x,2}]-2*x*D[y[x],x]+2*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
 

\[ y(x)\to -\frac {x^4}{6}-x^2+1 \]

✓ Sympy. Time used: 0.185 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + 2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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