84.32.8 problem 19.20 (a)

Internal problem ID [22316]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 19. Power series solutions about an ordinary point. Supplementary problems
Problem number : 19.20 (a)
Date solved : Thursday, October 02, 2025 at 08:37:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+x^{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 )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

Order:=6; 
ode:=diff(diff(y(x),x),x)-2*x*diff(y(x),x)+x^2*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 1-x -\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

ode=D[y[x],{x,2}]-2*x*D[y[x],x]+x^2*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {x^5}{20}-\frac {x^4}{12}-\frac {x^3}{3}-x+1 \]

✓ Sympy. Time used: 0.213 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {3 x^{5} r{\left (3 \right )}}{10} + C_{2} \left (- \frac {x^{6}}{45} - \frac {x^{4}}{12} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{20}\right ) + O\left (x^{6}\right ) \]