40.16.8 problem 15

Internal problem ID [6799]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 25. Integration in series. Supplemetary problems. Page 205
Problem number : 15
Date solved : Wednesday, March 05, 2025 at 02:46:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 34

Order:=6; 
ode:=diff(diff(y(x),x),x)-x*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 35

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

\[ y(x)\to c_1 \left (1-\frac {x^4}{12}\right )+c_2 \left (-\frac {x^5}{40}+\frac {x^3}{6}+x\right ) \]

✓ Sympy. Time used: 0.889 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {3 x^{5} r{\left (3 \right )}}{20} + C_{2} \left (- \frac {x^{6}}{90} - \frac {x^{4}}{12} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{20}\right ) + O\left (x^{6}\right ) \]